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SEO Keyword summary for en.wikipedia.org/wiki/inverse_matrix
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be mathbf
Focus keyword
Short and long tail
Short Tail Keywords mathbf matrix displaystyle |
long Tail Keywords (2 words) mathbf x mathrm d -1mathbf c d mathbf -1mathbf b |
long Tail Keywords (3 words) mathrm d mathbf mathbf a -1mathbf -mathbf ca -1mathbf d -mathbf ca mathbf a mathrm -mathbf bd -1mathbf ca -1mathbf b |
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invertible matrix wikipedia
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wikipedia free encyclopedia displaystyle mathbf mathbb lambda kmathbf mathrm dots cdots det vimathrm leq ijleq ujdelta frac detmathbf operatorname adj ntimes beginpmatrix endpmatrix tfrac textstyle leftbeginarraycccc endarrayright nmathbf xkxkaxk sum ssum ldots prod lklkloperatorname leftmathbf lrightkl klgeq lkln tll leftalright nsbnst tns iilambda leftlambda rightiifrac right over beginvmatrixmathbf endvmatrixmathbf endvmatrixbeginpmatrixmathbf vdots ddots ncdots nnendpmatrix rightij endvmatrixleftmathbf jiright beginbmatrixabcdendbmatrix beginbmatrixdbcaendbmatrixfrac adbcbeginbmatrixdbcaendbmatrix leftleftoperatorname rightmathbf beginbmatrixabcdefghiendbmatrix beginbmatrixabcdefghiendbmatrixmathrm beginbmatrixadgbehcfiendbmatrix beginalignedat aeifhquad dbichquad gbfcebdifgquad eaicgquad hafcdcdhegquad fahbgquad iaebdendalignedat aabbcc leftfrac leftoperatorname beginbmatrixmathbf endbmatrix times cdot beginbmatrixleftmathbf quad beginalignedleftmathbf endaligned endbmatrixbeginbmatrixmathbf tmathbf lim nto infty nprod lright rightn rightnmathbf leftxijright ixijmathbf imathbf icdot jdelta ixjimathbf wedge iwedge ncdot xji jrightleftdelta ijrightmathbf delta leftleftmathbf krightleftmathbf jcdot krightrightleftmathbf varepsilon mathcal ovarepsilon fmathbf tsum igimathbf thimathbf himathbf oleftvarepsilon beginalignedfrac nmathrm nifrac ifrac iendaligned beginalignedmathbf nvarepsilon nimathcal imathcal rightendaligned three dimensional euclidean space icon wikimedia foundation powered mediawiki
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SERP Title
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SERP Description
Domain Level SEO
Domain name
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Path name
inverse found in path !
matrix found in path !
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httpsenwikipediaorgwindexphptitleinvertiblematrixoldid1247727659
page information
printable version
inverse matrix
aiken computation laboratory
|
wiki read
linear algebra
square matrix
identity
multiplication
multiplicative inverse algorithm
field
if and only if
determinant
number line
complex plane
probability
almost never
right inverse
real
complex
algebraic structure
addition
multiplication
rings
commutative ring
noncommutative ring
group
general linear group
transpose
columnequivalent
pivot positions
kernel
injective
surjective
basis
unit
eigenvalue
elementary
moorepenrose inverse
orthonormal
dot product
orthogonal
gramschmidt process
involutory
adjugate
associativity
subset
null set
lebesgue measure
continuous function
polynomial
measure theory
almost all
open
dense
topological space
closed
nowhere dense
numerical calculations
illconditioned
necessary and sufficient condition
gaussian elimination
augmented
newtons method
victor pan
john reif
families
homotopy
matrix square root
imperfect computer arithmetic
cayleyhamilton method
trace
main diagonal
diophantine equation
bell polynomials
eigendecomposition of a matrix
eigenvector
diagonal
orthogonal
cholesky decomposition
positive definite
lower triangular
conjugate transpose
cramers rule
matrix of cofactors
recursive
reciprocal
computationally efficient
scalar
rule of sarrus
cross product
triple product
parallelepiped
block
schur complement
geodetic
tadeusz banachiewicz
nullity theorem
woodbury matrix identity
binomial inverse theorem
weinsteinaronszajn identity
zero
definite
divide and conquer algorithm
matrix multiplication algorithm
research into matrix multiplication complexity
neumann series
preconditioner
geometric sum
integer
rational
arbitraryprecision
padic
reciprocal basis
einstein summation
orthonormal basis
clifford algebra
geometric algebra
kronecker delta
generalized inverses
linear equations
lu decomposition
computer graphics
3d graphics
ray casting
mimo
wireless communications
frequency band
linear combination
decomposition
minor
partial inverse of a matrix
pseudoinverse
rybicki press algorithm
singular value decomposition
axler sheldon
undergraduate texts in mathematics
springer publishing
isbn
cambridge university press
bibcode
s2cid
ran raz
roman stephen
graduate texts in mathematics
encyclopedia of mathematics
ems press
cormen thomas h
leiserson charles e
rivest ronald l
stein clifford
introduction to algorithms
google books
youtube
mit opencourseware
outline
glossary
vector
vector space
scalar multiplication
vector projection
linear span
linear map
projection
linear independence
multilinear map
basis
change of basis
row and column vectors
row and column spaces
eigenvalues or eigenvectors
matrix
transformation
bilinear
orthogonality
hadamard product
inner product space
outer product
kronecker product
multilinear algebra
sevendimensional cross product
exterior algebra
bivector
multivector
tensor
outermorphism
dual
direct sum
function space
quotient
subspace
tensor product
numerical
floatingpoint
numerical stability
basic linear algebra subprograms
sparse
comparison of linear algebra libraries
alternant
antidiagonal
skewhermitian
skewsymmetric
arrowhead
band
bidiagonal
bisymmetric
blockdiagonal
block tridiagonal
boolean
cauchy
centrosymmetric
conference
complex hadamard
copositive
diagonally dominant
discrete fourier transform
equivalent
frobenius
generalized permutation
hadamard
hankel
hermitian
hessenberg
hollow
integer
logical
matrix unit
metzler
moore
nonnegative
pentadiagonal
permutation
persymmetric
polynomial
quaternionic
signature
skyline
sylvester
symmetric
toeplitz
triangular
tridiagonal
vandermonde
walsh
exchange
hilbert
lehmer
of ones
pascal
pauli
redheffer
shift
companion
convergent
defective
diagonalizable
hurwitz
positivedefinite
stieltjes
products
inverses
congruent
idempotent
nilpotent
normal
unimodular
unipotent
unitary
totally unimodular
weighing
alternating sign
bzout
carleman
cartan
circulant
cofactor
commutation
confusion
coxeter
distance
duplication and elimination
euclidean distance
fundamental linear differential equation
generator
gram
hessian
householder
jacobian
moment
payoff
pick
random
rotation
seifert
shear
similarity
symplectic
totally positive
statistics
centering
correlation
covariance
design
doubly stochastic
fisher information
precision
transition
graph theory
adjacency
biadjacency
degree
edmonds
incidence
laplacian
seidel adjacency
tutte
cabibbokobayashimaskawa
density
fundamental computer vision
fuzzy associative
gamma
gellmann
hamiltonian
irregular
overlap
state transition
substitution
z chemistry
jordan normal form
matrix exponential
matrix representation of conic sections
perfect matrix
row echelon form
wronskian
list of matrices
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b96b7261aa0889daebc9be22f4d9217116f3f3fb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{n}\mathbf {e} _{n-1}\cdots \mathbf {e} _{2}\mathbf {e} _{1}\mathbf {a} =\mathbf {i} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b0de28471a737cec8b509e29f4fa204a901c1984 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f17cb67790bab0a7611b951d1c22f52d633f2233 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{n}\mathbf {e} _{n-1}\cdots \mathbf {e} _{2}\mathbf {e} _{1}\mathbf {i} =\mathbf {i} \mathbf {a} ^{-1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf35037336b403f79c669259b85f00882b43422 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} \mathbf {a} ^{-1}=\mathbf {a} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3957b283573b8268471e2c9d12087ad80b905c91 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{n}\mathbf {e} _{n-1}\cdots \mathbf {e} _{2}\mathbf {e} _{1}\mathbf {i} ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9cedd88e295dc5669a444eaa8b7d7f4eb0a170 height: height attribute not set width: width attribute not set description: {\displaystyle x_{k+1}=2x_{k}-x_{k}ax_{k}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/10c3357d8c84994be3bd3d17459de4dca4a6e659 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det(\mathbf {a} )}}\sum _{s=0}^{n-1}\mathbf {a} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(\mathbf {a} ^{l}\right)^{k_{l}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7bd62dc750ae2faa00048f7345ba0b630f3df7 height: height attribute not set width: width attribute not set description: {\displaystyle k_{l}\geq 0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f278839d8c2666ae993aaecb2a902a6dfc1a0a1 height: height attribute not set width: width attribute not set description: {\displaystyle s+\sum _{l=1}^{n-1}lk_{l}=n-1.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba0f218c0326dda114533880792ef3d53999b3a height: height attribute not set width: width attribute not set description: {\displaystyle t_{l}=-(l-1)!\operatorname {tr} \left(a^{l}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/512f72f6bed764bb9fd2cbf4bae22bf9f1a03ef3 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det(\mathbf {a} )}}\sum _{s=1}^{n}\mathbf {a} ^{s-1}{\frac {(-1)^{n-1}}{(n-s)!}}b_{n-s}(t_{1},t_{2},\ldots ,t_{n-s}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e6cf923f9e79652a78ccb41acf6738779010697 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}=\mathbf {q} \mathbf {\lambda } ^{-1}\mathbf {q} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6 height: height attribute not set width: width attribute not set description: {\displaystyle q_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ed819091a61b8fab4dddd43de0684b0d56eb0597 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{ii}=\lambda _{i}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6c4635baf145f3d7a47466da26c888aa8ed1ca height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {q} ^{-1}=\mathbf {q} ^{\mathrm {t} }.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/839af615243151e8666a32caa26ab67108145c32 height: height attribute not set width: width attribute not set description: {\displaystyle \left[\lambda ^{-1}\right]_{ii}={\frac {1}{\lambda _{i}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2470a040625b308655f32373a99f401447d72452 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}=\left(\mathbf {l} ^{*}\right)^{-1}\mathbf {l} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/da69ff393cc3515fa4e17ea39779b71a94b19a92 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={1 \over {\begin{vmatrix}\mathbf {a} \end{vmatrix}}}\mathbf {c} ^{\mathrm {t} }={1 \over {\begin{vmatrix}\mathbf {a} \end{vmatrix}}}{\begin{pmatrix}\mathbf {c} _{11}&\mathbf {c} _{21}&\cdots &\mathbf {c} _{n1}\\\mathbf {c} _{12}&\mathbf {c} _{22}&\cdots &\mathbf {c} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {c} _{1n}&\mathbf {c} _{2n}&\cdots &\mathbf {c} _{nn}\\\end{pmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/16ce7e2c3c9b820ddbb6a2feb9a55601911862af height: height attribute not set width: width attribute not set description: {\displaystyle \left(\mathbf {a} ^{-1}\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {a} \end{vmatrix}}}\left(\mathbf {c} ^{\mathrm {t} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {a} \end{vmatrix}}}\left(\mathbf {c} _{ji}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed952db96b47e19ed879e28f83361155000b429 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{\det \mathbf {a} }}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a9c4572ae4bc8d969f0b14145493efd4791a32 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det \mathbf {a} }}\left[\left(\operatorname {tr} \mathbf {a} \right)\mathbf {i} -\mathbf {a} \right].} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3c15aec39a5ad87acba97400d8fd215f30f7b4 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {a} )}}{\begin{bmatrix}\,a&\,b&\,c\\\,d&\,e&\,f\\\,g&\,h&\,i\\\end{bmatrix}}^{\mathrm {t} }={\frac {1}{\det(\mathbf {a} )}}{\begin{bmatrix}\,a&\,d&\,g\\\,b&\,e&\,h\\\,c&\,f&\,i\\\end{bmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4d53623d8e3436da9b76bcfdd3ddf98cb935f4 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{alignedat}{6}a&={}&(ei-fh),&\quad &d&={}&-(bi-ch),&\quad &g&={}&(bf-ce),\\b&={}&-(di-fg),&\quad &e&={}&(ai-cg),&\quad &h&={}&-(af-cd),\\c&={}&(dh-eg),&\quad &f&={}&-(ah-bg),&\quad &i&={}&(ae-bd).\\\end{alignedat}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8d36697eaa2961b54cfc06e44c102f6dce854e97 height: height attribute not set width: width attribute not set description: {\displaystyle \det(\mathbf {a} )=aa+bb+cc.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0001e0f1d1cb3d6c54511dc5d0368202dee33b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det(\mathbf {a} )}}\left({\frac {1}{2}}\left[(\operatorname {tr} \mathbf {a} )^{2}-\operatorname {tr} (\mathbf {a} ^{2})\right]\mathbf {i} -\mathbf {a} \operatorname {tr} \mathbf {a} +\mathbf {a} ^{2}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1cfa670e35356eb168072436d5d94a40458a2f height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ={\begin{bmatrix}\mathbf {x} _{0}&\mathbf {x} _{1}&\mathbf {x} _{2}\end{bmatrix}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/799c59f89751f24a2719c4da95f1acdd3e2faf52 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a02b3163e7a24d90f15c809c2dca63ba5cccca7c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1274d8a5c59d33efd3639a4137c7f91598038d1b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77da2e7a835f573652de57c6d15ac04a83d2dddd height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det(\mathbf {a} )}}{\begin{bmatrix}{(\mathbf {x} _{1}\times \mathbf {x} _{2})}^{\mathrm {t} }\\{(\mathbf {x} _{2}\times \mathbf {x} _{0})}^{\mathrm {t} }\\{(\mathbf {x} _{0}\times \mathbf {x} _{1})}^{\mathrm {t} }\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/30be526c6fe4905526f4c094b1cee1b331606f35 height: height attribute not set width: width attribute not set description: {\displaystyle \det(\mathbf {a} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8853751ababdee69b55e57fab7c4bb19c0a259bb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} =\mathbf {a} ^{-1}\mathbf {a} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/727e0ffc544eb168334c361c9ce0ed4569e6ca9d height: height attribute not set width: width attribute not set description: {\displaystyle \det(\mathbf {a} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7c7671f1ebf97636abf149677369fe0644dd0e height: height attribute not set width: width attribute not set description: {\displaystyle 1={\frac {1}{\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}}\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/00e3f3b73d1714da529cd3c7ce1be0901b4d5bb2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}={\frac {1}{\det(\mathbf {a} )}}\left({\frac {1}{6}}\left[(\operatorname {tr} \mathbf {a} )^{3}-3\operatorname {tr} \mathbf {a} \operatorname {tr} (\mathbf {a} ^{2})+2\operatorname {tr} (\mathbf {a} ^{3})\right]\mathbf {i} -{\frac {1}{2}}\mathbf {a} \left[(\operatorname {tr} \mathbf {a} )^{2}-\operatorname {tr} (\mathbf {a} ^{2})\right]+\mathbf {a} ^{2}\operatorname {tr} \mathbf {a} -\mathbf {a} ^{3}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f3b9c79ecbaeb937d4b6101e6e7d4bc7e2ff13 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}\mathbf {a} &\mathbf {b} \\\mathbf {c} &\mathbf {d} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {a} ^{-1}+\mathbf {a} ^{-1}\mathbf {b} \left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\mathbf {ca} ^{-1}&-\mathbf {a} ^{-1}\mathbf {b} \left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\\-\left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\mathbf {ca} ^{-1}&\left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\end{bmatrix}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/025b91c0b7118b6093d336011d0be98efc06cd0c height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}\mathbf {a} &\mathbf {b} \\\mathbf {c} &\mathbf {d} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}&-\left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}\mathbf {bd} ^{-1}\\-\mathbf {d} ^{-1}\mathbf {c} \left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}&\quad \mathbf {d} ^{-1}+\mathbf {d} ^{-1}\mathbf {c} \left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}\mathbf {bd} ^{-1}\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d95307c24089c9a92aeec2960a79615f799382ca height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}&=\mathbf {a} ^{-1}+\mathbf {a} ^{-1}\mathbf {b} \left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\mathbf {ca} ^{-1}\\\left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}\mathbf {bd} ^{-1}&=\mathbf {a} ^{-1}\mathbf {b} \left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\\\mathbf {d} ^{-1}\mathbf {c} \left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}&=\left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\mathbf {ca} ^{-1}\\\mathbf {d} ^{-1}+\mathbf {d} ^{-1}\mathbf {c} \left(\mathbf {a} -\mathbf {bd} ^{-1}\mathbf {c} \right)^{-1}\mathbf {bd} ^{-1}&=\left(\mathbf {d} -\mathbf {ca} ^{-1}\mathbf {b} \right)^{-1}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/46798b30007cd793ef614505911e1400ac163dd6 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}\mathbf {a} &\mathbf {b} \\\mathbf {c} &\mathbf {d} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {a} -\mathbf {b} \mathbf {d} ^{-1}\mathbf {c} \right)^{-1}&\mathbf {0} \\\mathbf {0} &\left(\mathbf {d} -\mathbf {c} \mathbf {a} ^{-1}\mathbf {b} \right)^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {i} &-\mathbf {b} \mathbf {d} ^{-1}\\-\mathbf {c} \mathbf {a} ^{-1}&\mathbf {i} \end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f46b724dcbc889e39cb231017ec1c4c856fd3d52 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}\mathbf {a} &\mathbf {0} \\\mathbf {c} &\mathbf {d} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {a} ^{-1}&\mathbf {0} \\-\mathbf {d} ^{-1}\mathbf {ca} ^{-1}&\mathbf {d} ^{-1}\end{bmatrix}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c764c2ff7c3427a539e747a60f355e66df1f494c height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{bmatrix}\mathbf {a} &\mathbf {c} ^{t}\\\mathbf {c} &\mathbf {d} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {a} ^{-1}+\mathbf {a} ^{-1}\mathbf {c} ^{t}\mathbf {s} ^{-1}\mathbf {c} \mathbf {a} ^{-1}&-\mathbf {a} ^{-1}\mathbf {c} ^{t}\mathbf {s} ^{-1}\\-\mathbf {s} ^{-1}\mathbf {c} \mathbf {a} ^{-1}&\mathbf {s} ^{-1}\end{bmatrix}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/116040b2578aac93c76707f765c77e296dbc4b62 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {s} =\mathbf {d} -\mathbf {c} \mathbf {a} ^{-1}\mathbf {c} ^{t}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5995575259860d197d5feaeb7ca5269a924ce1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {w} _{1}=\mathbf {c} \mathbf {a} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b492832cce63627e9a1493015df7fcbd2a8229c1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {w} _{2}=\mathbf {w} _{1}\mathbf {c} ^{t}=\mathbf {c} \mathbf {a} ^{-1}\mathbf {c} ^{t},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1e879ba28ff837453dccaa9244f90a0cfa9290 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {w} _{3}=\mathbf {s} ^{-1}\mathbf {w} _{1}=\mathbf {s} ^{-1}\mathbf {c} \mathbf {a} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d49e2504b21309dba30be347cd49a2be68b01f91 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {w} _{4}=\mathbf {w} _{1}^{t}\mathbf {w} _{3}=\mathbf {a} ^{-1}\mathbf {c} ^{t}\mathbf {s} ^{-1}\mathbf {c} \mathbf {a} ^{-1},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e499ae5946af9c09777ada933051b3669d3372c2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {m} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c2eed65f986c38e0e27258769326492ba5f7033e height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {m} ^{t}\mathbf {m} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc843fd05b4f5be9fee4464bb1b227272576255 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {m} ^{-1}=\left(\mathbf {m} ^{t}\mathbf {m} \right)^{-1}\mathbf {m} ^{t}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4049cb0599635cbd510f7b6b359fc849acb5c34b height: height attribute not set width: width attribute not set description: {\displaystyle \lim _{n\to \infty }(\mathbf {i} -\mathbf {a} )^{n}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/330be98a0681c50d5f799130115ee6cd38c3fa9d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}=\sum _{n=0}^{\infty }(\mathbf {i} -\mathbf {a} )^{n}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5af1a7ba7738e8870dfc0f47272cbe297adbdf86 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{n=0}^{2^{l}-1}(\mathbf {i} -\mathbf {a} )^{n}=\prod _{l=0}^{l-1}\left(\mathbf {i} +(\mathbf {i} -\mathbf {a} )^{2^{l}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d573ded3c7edd0acf6dd172261049a60b605f1 height: height attribute not set width: width attribute not set description: {\displaystyle \lim _{n\to \infty }\left(\mathbf {i} -\mathbf {x} ^{-1}\mathbf {a} \right)^{n}=0\mathrm {~~or~~} \lim _{n\to \infty }\left(\mathbf {i} -\mathbf {a} \mathbf {x} ^{-1}\right)^{n}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ad7119b544c3d8d9076c3303aa863657ee2a8 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}=\sum _{n=0}^{\infty }\left(\mathbf {x} ^{-1}(\mathbf {x} -\mathbf {a} )\right)^{n}\mathbf {x} ^{-1}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/66d0a8de173ec0db7487f1ff2446500075e67d86 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}=\mathbf {x} ^{-1}-{\frac {\mathbf {x} ^{-1}(\mathbf {a} -\mathbf {x} )\mathbf {x} ^{-1}}{1+\operatorname {tr} \left(\mathbf {x} ^{-1}(\mathbf {a} -\mathbf {x} )\right)}}~.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcdbd40b9ad9e18e49f3dbee92a5f67de48c667 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} =\left[x^{ij}\right]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a261efc4a9aa731385da988d0ccb68a202d522ea height: height attribute not set width: width attribute not set description: {\displaystyle 1\leq i,j\leq n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/132d7b2e8170903bb971940ea55e8c45c951f8e7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{i}=x^{ij}\mathbf {e} _{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7742f3851d608848056eab437b32e8b753dd5b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/51e31f5007a1be571e45133d5fc8a318d74202fd height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{i}=\mathbf {e} ^{i},\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f763c639cd1ef6bcbaffd32c3f3659b8daf077 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{i}=x_{ji}\mathbf {e} ^{j}=(-1)^{i-1}(\mathbf {x} _{1}\wedge \cdots \wedge ()_{i}\wedge \cdots \wedge \mathbf {x} _{n})\cdot (\mathbf {x} _{1}\wedge \ \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})^{-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/78d43f884ab940a905153b017d09490e0ed6a6e9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{-1}=[x_{ji}].} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2d567afc79f80afc8766df857153dcbd131090 height: height attribute not set width: width attribute not set description: {\displaystyle ()_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9a7653709fa1f5ff7818e182d141f5c09d7559 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b784769b878e5e00d9c003037c0e0aa44dd292 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} \mathbf {x} ^{-1}=\left[\mathbf {x} _{i}\cdot \mathbf {x} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {i} _{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff8a003e21c4cc60d2577b44edecf710120992b height: height attribute not set width: width attribute not set description: {\displaystyle \delta _{i}^{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b3a1a25ab9213ccc1996b44c9dbc688c4b4068 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{-1}\mathbf {x} =\left[\left(\mathbf {e} _{i}\cdot \mathbf {x} ^{k}\right)\left(\mathbf {e} ^{j}\cdot \mathbf {x} _{k}\right)\right]=\left[\mathbf {e} _{i}\cdot \mathbf {e} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {i} _{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c423b4b8de32bddfe056fff2e1e879d58f56527 height: height attribute not set width: width attribute not set description: {\displaystyle (\mathbf {x} _{1}\wedge \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f1a7661254642247ea754982e6649dbade6550 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\mathrm {d} \mathbf {a} ^{-1}}{\mathrm {d} t}}=-\mathbf {a} ^{-1}{\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}\mathbf {a} ^{-1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9466de8b4f79d93ee74e2b3d6dd52e2260c901c3 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}\mathbf {a} =\mathbf {i} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/689280edcad0d944ff49fabb86ed109f942dd7a4 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\mathrm {d} (\mathbf {a} ^{-1}\mathbf {a} )}{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {a} ^{-1}}{\mathrm {d} t}}\mathbf {a} +\mathbf {a} ^{-1}{\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {i} }{\mathrm {d} t}}=\mathbf {0} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/167b6b416b51d626642c11c1796e62060a40e36c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}{\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4a37527a4700b3ad51e31a1fba1dbfc2a5af22 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ^{-1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6065bd95f24e872c3385074f2d2885681fe38d24 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\mathbf {a} +\varepsilon \mathbf {x} \right)^{-1}=\mathbf {a} ^{-1}-\varepsilon \mathbf {a} ^{-1}\mathbf {x} \mathbf {a} ^{-1}+{\mathcal {o}}(\varepsilon ^{2})\,.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/95ff9e5f6a8ec5bace272992c31da3e6428a6d2a height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\mathrm {d} f(\mathbf {a} )}{\mathrm {d} t}}=\sum _{i}g_{i}(\mathbf {a} ){\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}h_{i}(\mathbf {a} ),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1671ff10b475d7644272cfb4b0226c489172ca height: height attribute not set width: width attribute not set description: {\displaystyle f(\mathbf {a} +\varepsilon \mathbf {x} )=f(\mathbf {a} )+\varepsilon \sum _{i}g_{i}(\mathbf {a} )\mathbf {x} h_{i}(\mathbf {a} )+{\mathcal {o}}\left(\varepsilon ^{2}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b height: height attribute not set width: width attribute not set description: {\displaystyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23147ed8dd799592c46c9a55e2447a8eeb730fef height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {a} ^{n}}{\mathrm {d} t}}&=\sum _{i=1}^{n}\mathbf {a} ^{i-1}{\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}\mathbf {a} ^{n-i},\\{\frac {\mathrm {d} \mathbf {a} ^{-n}}{\mathrm {d} t}}&=-\sum _{i=1}^{n}\mathbf {a} ^{-i}{\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}\mathbf {a} ^{-(n+1-i)}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5eaeba69cd30b9c5c6850fc751550b86a99fe033 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}(\mathbf {a} +\varepsilon \mathbf {x} )^{n}&=\mathbf {a} ^{n}+\varepsilon \sum _{i=1}^{n}\mathbf {a} ^{i-1}\mathbf {x} \mathbf {a} ^{n-i}+{\mathcal {o}}\left(\varepsilon ^{2}\right),\\(\mathbf {a} +\varepsilon \mathbf {x} )^{-n}&=\mathbf {a} ^{-n}-\varepsilon \sum _{i=1}^{n}\mathbf {a} ^{-i}\mathbf {x} \mathbf {a} ^{-(n+1-i)}+{\mathcal {o}}\left(\varepsilon ^{2}\right).\end{aligned}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/2/2f/linear_subspaces_with_shading.svg/80px-linear_subspaces_with_shading.svg.png height: 58 width: 80 description: three dimensional euclidean space |
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https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/nuvola_apps_edu_mathematics_blue-p.svg/16px-nuvola_apps_edu_mathematics_blue-p.svg.png height: 16 width: 16 description: icon |
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https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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http://en.wikipedia.org/w/resources/assets/poweredby_mediawiki.svg height: 31 width: 88 description: powered by mediawiki |
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