Invertible matrix

Inlinear algebra,ann-by-nsquare matrixAis calledinvertible(alsononsingular,nondegenerateor rarelyregular) if there exists ann-by-nsquare matrixBsuch that$$\mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},$$whereI_ndenotes then-by-nidentity matrixand the multiplication used is ordinarymatrix multiplication.If this is the case, then the matrixBis uniquely determined byA,and is called the(multiplicative)inverseofA,denoted byA⁻¹.Matrix inversionis the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.

Over afield,a square matrix that isnotinvertible is calledsingularordegenerate.A square matrix with entries in a field is singularif and only ifitsdeterminantis zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on thenumber lineorcomplex plane,theprobabilitythat the matrix is singular is 0, that is, it will"almost never"be singular. Non-square matrices, i.e.m-by-nmatrices for whichm≠n,do not have an inverse. However, in some cases such a matrix may have aleft inverseorright inverse.IfAism-by-nand therankofAis equal ton,(n≤m), thenAhas a left inverse, ann-by-mmatrixBsuch thatBA=I_n.IfAhas rankm(m≤n), then it has a right inverse, ann-by-mmatrixBsuch thatAB=I_m.

While the most common case is that of matrices over therealorcomplexnumbers, all these definitions can be given for matrices over anyalgebraic structureequipped withadditionandmultiplication(i.e.rings). However, in the case of a ring beingcommutative,the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For anoncommutative ring,the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.

The set ofn×ninvertible matrices together with the operation ofmatrix multiplicationand entries from ringRform agroup,thegeneral linear groupof degreen,denotedGL_n(R).

Properties

The invertible matrix theorem

LetAbe a squaren-by-nmatrix over afieldK(e.g., the field⁠$\mathbb {R}$⁠of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:

• Ais invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists aBsuch thatAB=I_n=BA.(In this statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible", in which one-sided inverses are considered.)
• The linear transformation mappingxtoAxis invertible, i.e., has an inverse under function composition. (Here, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible" )
• ThetransposeA^Tis an invertible matrix.
• Aisrow-equivalentto then-by-nidentity matrixI_n.
• Aiscolumn-equivalentto then-by-nidentity matrixI_n.
• Ahasnpivot positions.
• Ahas fullrank:rankA=n.
• Ahas a trivialkernel:ker(A) = {0}.
• The linear transformation mappingxtoAxis bijective; that is, the equationAx=bhas exactly one solution for eachbinKⁿ.(Here, "bijective" can equivalently be replaced with "injective"or"surjective")
• The columns ofAform abasisofKⁿ.(In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set" )
• The rows ofAform a basis ofKⁿ.(Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set" )
• ThedeterminantofAis nonzero:detA≠ 0.(In general, a square matrix over acommutative ringis invertible if and only if its determinant is aunit(i.e. multiplicatively invertible element) of that ring.
• The number 0 is not aneigenvalueofA.(More generally, a number$\lambda$is an eigenvalue ofAif the matrix$\mathbf {A} -\lambda \mathbf {I}$is singular, whereIis the identity matrix.)
• The matrixAcan be expressed as a finite product ofelementary matrices.

Other properties

Furthermore, the following properties hold for an invertible matrixA:

• $(\mathbf {A} ^{-1})^{-1}=\mathbf {A}$
• $(k\mathbf {A} )^{-1}=k^{-1}\mathbf {A} ^{-1}$for nonzero scalark
• $(\mathbf {Ax} )^{+}=\mathbf {x} ^{+}\mathbf {A} ^{-1}$ifAhas orthonormal columns, where⁺denotes theMoore–Penrose inverseandxis a vector
• $(\mathbf {A} ^{\mathrm {T} })^{-1}=(\mathbf {A} ^{-1})^{\mathrm {T} }$
• For any invertiblen-by-nmatricesAandB,$(\mathbf {AB} )^{-1}=\mathbf {B} ^{-1}\mathbf {A} ^{-1}.$More generally, if$\mathbf {A} _{1},\dots,\mathbf {A} _{k}$are invertiblen-by-nmatrices, then$(\mathbf {A} _{1}\mathbf {A} _{2}\cdots \mathbf {A} _{k-1}\mathbf {A} _{k})^{-1}=\mathbf {A} _{k}^{-1}\mathbf {A} _{k-1}^{-1}\cdots \mathbf {A} _{2}^{-1}\mathbf {A} _{1}^{-1}.$
• $\det \mathbf {A} ^{-1}=(\det \mathbf {A} )^{-1}.$

The rows of the inverse matrixVof a matrixUareorthonormalto the columns ofU(and vice versa interchanging rows for columns). To see this, suppose thatUV=VU=Iwhere the rows ofVare denoted as$v_{i}^{\mathrm {T} }$and the columns ofUas$u_{j}$for$1\leq i,j\leq n.$Then clearly, theEuclidean inner productof any two$v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.$This property can also be useful in constructing the inverse of a square matrix in some instances, where a set oforthogonalvectors (but not necessarily orthonormal vectors) to the columns ofUare known. In which case, one can apply the iterativeGram–Schmidt processto this initial set to determine the rows of the inverseV.

A matrix that is its own inverse (i.e., a matrixAsuch thatA=A⁻¹,and consequentlyA²=I), is called aninvolutory matrix.

In relation to its adjugate

Theadjugateof a matrixAcan be used to find the inverse ofAas follows:

IfAis an invertible matrix, then

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\operatorname {adj} (\mathbf {A} ).$

In relation to the identity matrix

It follows from theassociativityof matrix multiplication that if

$\mathbf {AB} =\mathbf {I} \$

forfinite squarematricesAandB,then also

$\mathbf {BA} =\mathbf {I} \$

Density

Over the field of real numbers, the set of singularn-by-nmatrices, considered as asubsetof⁠$\mathbb {R} ^{n\times n},$⁠is anull set,that is, hasLebesguemeasure zero.This is true because singular matrices are the roots of thedeterminantfunction. This is acontinuous functionbecause it is apolynomialin the entries of the matrix. Thus in the language ofmeasure theory,almost alln-by-nmatrices are invertible.

Furthermore, then-by-ninvertible matrices are adenseopen setin thetopological spaceof alln-by-nmatrices. Equivalently, the set of singular matrices isclosedandnowhere densein the space ofn-by-nmatrices.

In practice however, one may encounter non-invertible matrices. And innumerical calculations,matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to beill-conditioned.

Examples

An example with rank ofn− 1is a non-invertible matrix

$\mathbf {A} ={\begin{pmatrix}2&4\\2&4\end{pmatrix}}.$

We can see the rank of this 2-by-2 matrix is 1, which isn− 1 ≠n,so it is non-invertible.

Consider the following 2-by-2 matrix:

$\mathbf {B} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.$

The matrix$\mathbf {B}$is invertible. To check this, one can compute that${\textstyle \det \mathbf {B} =-{\frac {1}{2}}}$,which is non-zero.

As an example of a non-invertible, or singular, matrix, consider the matrix

$\mathbf {C} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\{\tfrac {2}{3}}&-1\end{pmatrix}}.$

The determinant of$\mathbf {C}$is 0, which is anecessary and sufficient conditionfor a matrix to be non-invertible.

Methods of matrix inversion

Gaussian elimination

Gaussian eliminationis a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, anaugmented matrixis first created with the left side being the matrix to invert and the right side being theidentity matrix.Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.

For example, take the following matrix:$\mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.$

The first step to compute its inverse is to create the augmented matrix$\left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\1&-1&0&1\end{array}}\right).$

Call the first row of this matrix$R_{1}$and the second row$R_{2}$.Then, add row 1 to row 2$(R_{1}+R_{2}\to R_{2}).$This yields$\left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).$

Next, subtract row 2, multiplied by 3, from row 1$(R_{1}-3\,R_{2}\to R_{1}),$which yields$\left({\begin{array}{cc|cc}-1&0&-2&-3\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).$

Finally, multiply row 1 by −1$(-R_{1}\to R_{1})$and row 2 by 2$(2\,R_{2}\to R_{2}).$This yields the identity matrix on the left side and the inverse matrix on the right:$\left({\begin{array}{cc|cc}1&0&2&3\\0&1&2&2\end{array}}\right).$

Thus,$\mathbf {A} ^{-1}={\begin{pmatrix}2&3\\2&2\end{pmatrix}}.$

The reason it works is that the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations usingelementary matrices($\mathbf {E} _{n}$), such as$\mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {A} =\mathbf {I}.$

Applying right-multiplication using$\mathbf {A} ^{-1},$we get$\mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} =\mathbf {I} \mathbf {A} ^{-1}.$And the right side$\mathbf {I} \mathbf {A} ^{-1}=\mathbf {A} ^{-1},$which is the inverse we want.

To obtain$\mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I},$we create the augumented matrix by combiningAwithIand applyingGaussian elimination.The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomesI,the right portion applied the same elementary row operation sequence will becomeA⁻¹.

Newton's method

A generalization ofNewton's methodas used for amultiplicative inverse algorithmmay be convenient, if it is convenient to find a suitable starting seed:

$X_{k+1}=2X_{k}-X_{k}AX_{k}.$

Victor PanandJohn Reifhave done work that includes ways of generating a starting seed.

Newton's method is particularly useful when dealing withfamiliesof related matrices that behave enough like the sequence manufactured for thehomotopyabove: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtainingmatrix square roots by Denman–Beavers iteration;this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due toimperfect computer arithmetic.

Cayley–Hamilton method

TheCayley–Hamilton theoremallows the inverse ofAto be expressed in terms ofdet(A),traces and powers ofA:

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(\mathbf {A} ^{l}\right)^{k_{l}},$

wherenis size ofA,andtr(A)is thetraceof matrixAgiven by the sum of themain diagonal.The sum is taken oversand the sets of all$k_{l}\geq 0$satisfying the linearDiophantine equation

$s+\sum _{l=1}^{n-1}lk_{l}=n-1.$

The formula can be rewritten in terms of completeBell polynomialsof arguments$t_{l}=-(l-1)!\operatorname {tr} \left(A^{l}\right)$as

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=1}^{n}\mathbf {A} ^{s-1}{\frac {(-1)^{n-1}}{(n-s)!}}B_{n-s}(t_{1},t_{2},\ldots,t_{n-s}).$

This is described in more detail underCayley–Hamilton method.

Eigendecomposition

Main article:Eigendecomposition of a matrix

If matrixAcan be eigendecomposed, and if none of its eigenvalues are zero, thenAis invertible and its inverse is given by

$\mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1},$

whereQis the square(N×N)matrix whoseith column is theeigenvector$q_{i}$ofA,andΛis thediagonal matrixwhose diagonal entries are the corresponding eigenvalues, that is,$\Lambda _{ii}=\lambda _{i}.$If Ais symmetric,Qis guaranteed to be anorthogonal matrix,therefore$\mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }.$Furthermore, becauseΛis a diagonal matrix, its inverse is easy to calculate:

$\left[\Lambda ^{-1}\right]_{ii}={\frac {1}{\lambda _{i}}}.$

Cholesky decomposition

Main article:Cholesky decomposition

If matrixAispositive definite,then its inverse can be obtained as

$\mathbf {A} ^{-1}=\left(\mathbf {L} ^{*}\right)^{-1}\mathbf {L} ^{-1},$

whereLis thelower triangularCholesky decompositionofA,andL*denotes theconjugate transposeofL.

Analytic solution

Main article:Cramer's rule

Writing the transpose of thematrix of cofactors,known as anadjugate matrix,can also be an efficient way to calculate the inverse ofsmallmatrices, but thisrecursivemethod is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:

$\mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\mathbf {C} ^{\mathrm {T} }={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}{\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{21}&\cdots &\mathbf {C} _{n1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1n}&\mathbf {C} _{2n}&\cdots &\mathbf {C} _{nn}\\\end{pmatrix}}$

so that

$\left(\mathbf {A} ^{-1}\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} ^{\mathrm {T} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ji}\right)$

where|A|is thedeterminantofA,Cis thematrix of cofactors,andC^Trepresents the matrixtranspose.

Inversion of 2 × 2 matrices

Thecofactor equationlisted above yields the following result for2 × 2matrices. Inversion of these matrices can be done as follows:

$\mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{\det \mathbf {A} }}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}.$

This is possible because1/(ad−bc)is thereciprocalof the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.

The Cayley–Hamilton method gives

$\mathbf {A} ^{-1}={\frac {1}{\det \mathbf {A} }}\left[\left(\operatorname {tr} \mathbf {A} \right)\mathbf {I} -\mathbf {A} \right].$

Inversion of 3 × 3 matrices

Acomputationally efficient3 × 3matrix inversion is given by

$\mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,I\\\end{bmatrix}}^{\mathrm {T} }={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,I\\\end{bmatrix}}$

(where thescalarAis not to be confused with the matrixA).

If the determinant is non-zero, the matrix is invertible, with the entries of the intermediary matrix on the right side above given by

${\begin{alignedat}{6}A&={}&(ei-fh),&\quad &D&={}&-(bi-ch),&\quad &G&={}&(bf-ce),\\B&={}&-(di-fg),&\quad &E&={}&(ai-cg),&\quad &H&={}&-(af-cd),\\C&={}&(dh-eg),&\quad &F&={}&-(ah-bg),&\quad &I&={}&(ae-bd).\\\end{alignedat}}$

The determinant ofAcan be computed by applying therule of Sarrusas follows:

$\det(\mathbf {A} )=aA+bB+cC.$

The Cayley–Hamilton decomposition gives

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left({\frac {1}{2}}\left[(\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} (\mathbf {A} ^{2})\right]\mathbf {I} -\mathbf {A} \operatorname {tr} \mathbf {A} +\mathbf {A} ^{2}\right).$

The general3 × 3inverse can be expressed concisely in terms of thecross productandtriple product.If a matrix$\mathbf {A} ={\begin{bmatrix}\mathbf {x} _{0}&\mathbf {x} _{1}&\mathbf {x} _{2}\end{bmatrix}}$(consisting of three column vectors,$\mathbf {x} _{0}$,$\mathbf {x} _{1}$,and$\mathbf {x} _{2}$) is invertible, its inverse is given by

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}{(\mathbf {x} _{1}\times \mathbf {x} _{2})}^{\mathrm {T} }\\{(\mathbf {x} _{2}\times \mathbf {x} _{0})}^{\mathrm {T} }\\{(\mathbf {x} _{0}\times \mathbf {x} _{1})}^{\mathrm {T} }\end{bmatrix}}.$

The determinant ofA,det(A),is equal to the triple product ofx₀,x₁,andx₂—the volume of theparallelepipedformed by the rows or columns:

$\det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).$

The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row ofA^–1is orthogonal to the non-corresponding two columns ofA(causing the off-diagonal terms of$\mathbf {I} =\mathbf {A} ^{-1}\mathbf {A}$be zero). Dividing by

$\det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})$

causes the diagonal entries ofI=A⁻¹Ato be unity. For example, the first diagonal is:

$1={\frac {1}{\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}}\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).$

Inversion of 4 × 4 matrices

With increasing dimension, expressions for the inverse ofAget complicated. Forn= 4,the Cayley–Hamilton method leads to an expression that is still tractable:

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left({\frac {1}{6}}\left[(\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} (\mathbf {A} ^{2})+2\operatorname {tr} (\mathbf {A} ^{3})\right]\mathbf {I} -{\frac {1}{2}}\mathbf {A} \left[(\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} (\mathbf {A} ^{2})\right]+\mathbf {A} ^{2}\operatorname {tr} \mathbf {A} -\mathbf {A} ^{3}\right).$

Blockwise inversion

Matrices can also beinverted blockwiseby using the following analytic inversion formula:

${\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}&-\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\\-\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}&\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\end{bmatrix}},$ (1)

whereA,B,CandDarematrix sub-blocksof arbitrary size. (Amust be square, so that it can be inverted. Furthermore,AandD−CA⁻¹Bmust be nonsingular.) This strategy is particularly advantageous ifAis diagonal andD−CA⁻¹B(theSchur complementofA) is a small matrix, since they are the only matrices requiring inversion.

This technique was reinvented several times and is due toHans Boltz(1923),who used it for the inversion ofgeodeticmatrices, andTadeusz Banachiewicz(1937), who generalized it and proved its correctness.

Thenullity theoremsays that the nullity ofAequals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity ofBequals the nullity of the sub-block in the upper right of the inverse matrix.

The inversion procedure that led to Equation (1) performed matrix block operations that operated onCandDfirst. Instead, ifAandBare operated on first, and providedDandA−BD⁻¹Care nonsingular,the result is

${\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&-\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}\\-\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&\quad \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}\end{bmatrix}}.$ (2)

Equating Equations (1) and (2) leads to

${\begin{aligned}\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\mathbf {A} ^{-1}\mathbf {B} \left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\\\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\mathbf {CA} ^{-1}\\\mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} \left(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} \right)^{-1}\mathbf {BD} ^{-1}&=\left(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} \right)^{-1}\end{aligned}}$ (3)

where Equation (3) is theWoodbury matrix identity,which is equivalent to thebinomial inverse theorem.

IfAandDare both invertible, then the above two block matrix inverses can be combined to provide the simple factorization

${\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\left(\mathbf {A} -\mathbf {B} \mathbf {D} ^{-1}\mathbf {C} \right)^{-1}&\mathbf {0} \\\mathbf {0} &\left(\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {B} \right)^{-1}\end{bmatrix}}{\begin{bmatrix}\mathbf {I} &-\mathbf {B} \mathbf {D} ^{-1}\\-\mathbf {C} \mathbf {A} ^{-1}&\mathbf {I} \end{bmatrix}}.$ (2)

By theWeinstein–Aronszajn identity,one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

This formula simplifies significantly when the upper right block matrixBis thezero matrix.This formulation is useful when the matricesAandDhave relatively simple inverse formulas (orpseudo inversesin the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes

${\begin{bmatrix}\mathbf {A} &\mathbf {0} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}&\mathbf {0} \\-\mathbf {D} ^{-1}\mathbf {CA} ^{-1}&\mathbf {D} ^{-1}\end{bmatrix}}.$

If the given invertible matrix is a symmetric matrix with invertible blockAthe following block inverse formula holds

${\begin{bmatrix}\mathbf {A} &\mathbf {C} ^{T}\\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&-\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\\-\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1}&\mathbf {S} ^{-1}\end{bmatrix}},$ (4)

where$\mathbf {S} =\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T}$.This requires 2 inversions of the half-sized matricesAandSand only 4 multiplications of half-sized matrices, if organized properly$\mathbf {W} _{1}=\mathbf {C} \mathbf {A} ^{-1},$$\mathbf {W} _{2}=\mathbf {W} _{1}\mathbf {C} ^{T}=\mathbf {C} \mathbf {A} ^{-1}\mathbf {C} ^{T},$$\mathbf {W} _{3}=\mathbf {S} ^{-1}\mathbf {W} _{1}=\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},$$\mathbf {W} _{4}=\mathbf {W} _{1}^{T}\mathbf {W} _{3}=\mathbf {A} ^{-1}\mathbf {C} ^{T}\mathbf {S} ^{-1}\mathbf {C} \mathbf {A} ^{-1},$together with some additions, subtractions, negations and transpositions of negligible complexity. Any matrix$\mathbf {M}$has an associated positive semidefinite, symmetric matrix$\mathbf {M} ^{T}\mathbf {M}$,which is exactly invertible (and positive definite), if and only if$\mathbf {M}$is invertible. By writing$\mathbf {M} ^{-1}=\left(\mathbf {M} ^{T}\mathbf {M} \right)^{-1}\mathbf {M} ^{T}$matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because thepositive definite matrix$\mathbf {M} ^{T}\mathbf {M}$satisfies the invertibility condition for its left upper blockA.

These formulas together allow to construct adivide and conquer algorithmthat uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as thematrix multiplication algorithmthat is used internally.Research into matrix multiplication complexityshows that there exist matrix multiplication algorithms with a complexity ofO(n^2.371552)operations, while the best proven lower bound isΩ(n²logn).

By Neumann series

If a matrixAhas the property that

$\lim _{n\to \infty }(\mathbf {I} -\mathbf {A} )^{n}=0$

thenAis nonsingular and its inverse may be expressed by aNeumann series:

$\mathbf {A} ^{-1}=\sum _{n=0}^{\infty }(\mathbf {I} -\mathbf {A} )^{n}.$

Truncating the sum results in an "approximate" inverse which may be useful as apreconditioner.Note that a truncated series can be accelerated exponentially by noting that the Neumann series is ageometric sum.As such, it satisfies

$\sum _{n=0}^{2^{L}-1}(\mathbf {I} -\mathbf {A} )^{n}=\prod _{l=0}^{L-1}\left(\mathbf {I} +(\mathbf {I} -\mathbf {A} )^{2^{l}}\right)$.

Therefore, only2L− 2matrix multiplications are needed to compute2^Lterms of the sum.

More generally, ifAis "near" the invertible matrixXin the sense that

$\lim _{n\to \infty }\left(\mathbf {I} -\mathbf {X} ^{-1}\mathbf {A} \right)^{n}=0\mathrm {~~or~~} \lim _{n\to \infty }\left(\mathbf {I} -\mathbf {A} \mathbf {X} ^{-1}\right)^{n}=0$

thenAis nonsingular and its inverse is

$\mathbf {A} ^{-1}=\sum _{n=0}^{\infty }\left(\mathbf {X} ^{-1}(\mathbf {X} -\mathbf {A} )\right)^{n}\mathbf {X} ^{-1}~.$

If it is also the case thatA−Xhasrank1 then this simplifies to

$\mathbf {A} ^{-1}=\mathbf {X} ^{-1}-{\frac {\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\mathbf {X} ^{-1}}{1+\operatorname {tr} \left(\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\right)}}~.$

p-adic approximation

IfAis a matrix withintegerorrationalentries and we seek a solution inarbitrary-precisionrationals, then ap-adicapproximation method converges to an exact solution inO(n⁴log²n),assuming standardO(n³)matrix multiplication is used.The method relies on solvingnlinear systems via Dixon's method ofp-adic approximation (each inO(n³log²n)) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.

Reciprocal basis vectors method

Main article:Reciprocal basis

Given ann×nsquare matrix$\mathbf {X} =\left[x^{ij}\right]$,$1\leq i,j\leq n$,withnrows interpreted asnvectors$\mathbf {x} _{i}=x^{ij}\mathbf {e} _{j}$(Einstein summationassumed) where the$\mathbf {e} _{j}$are a standardorthonormal basisofEuclidean space$\mathbb {R} ^{n}$($\mathbf {e} _{i}=\mathbf {e} ^{i},\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}$), then usingClifford algebra(orgeometric algebra) we compute the reciprocal (sometimes calleddual) column vectors:

$\mathbf {x} ^{i}=x_{ji}\mathbf {e} ^{j}=(-1)^{i-1}(\mathbf {x} _{1}\wedge \cdots \wedge ()_{i}\wedge \cdots \wedge \mathbf {x} _{n})\cdot (\mathbf {x} _{1}\wedge \ \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})^{-1}$

as the columns of the inverse matrix$\mathbf {X} ^{-1}=[x_{ji}].$Note that, the place "$()_{i}$"indicates that"$\mathbf {x} _{i}$"is removed from that place in the above expression for$\mathbf {x} ^{i}$.We then have$\mathbf {X} \mathbf {X} ^{-1}=\left[\mathbf {x} _{i}\cdot \mathbf {x} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {I} _{n}$,where$\delta _{i}^{j}$is theKronecker delta.We also have$\mathbf {X} ^{-1}\mathbf {X} =\left[\left(\mathbf {e} _{i}\cdot \mathbf {x} ^{k}\right)\left(\mathbf {e} ^{j}\cdot \mathbf {x} _{k}\right)\right]=\left[\mathbf {e} _{i}\cdot \mathbf {e} ^{j}\right]=\left[\delta _{i}^{j}\right]=\mathbf {I} _{n}$,as required. If the vectors$\mathbf {x} _{i}$are not linearly independent, then$(\mathbf {x} _{1}\wedge \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})=0$and the matrix$\mathbf {X}$is not invertible (has no inverse).

Derivative of the matrix inverse

Suppose that the invertible matrixAdepends on a parametert.Then the derivative of the inverse ofAwith respect totis given by

${\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.$

To derive the above expression for the derivative of the inverse ofA,one can differentiate the definition of the matrix inverse$\mathbf {A} ^{-1}\mathbf {A} =\mathbf {I}$and then solve for the inverse ofA:

${\frac {\mathrm {d} (\mathbf {A} ^{-1}\mathbf {A} )}{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}\mathbf {A} +\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {I} }{\mathrm {d} t}}=\mathbf {0}.$

Subtracting$\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}$from both sides of the above and multiplying on the right by$\mathbf {A} ^{-1}$gives the correct expression for the derivative of the inverse:

${\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.$

Similarly, if$\varepsilon$is a small number then

$\left(\mathbf {A} +\varepsilon \mathbf {X} \right)^{-1}=\mathbf {A} ^{-1}-\varepsilon \mathbf {A} ^{-1}\mathbf {X} \mathbf {A} ^{-1}+{\mathcal {O}}(\varepsilon ^{2})\,.$

More generally, if

${\frac {\mathrm {d} f(\mathbf {A} )}{\mathrm {d} t}}=\sum _{i}g_{i}(\mathbf {A} ){\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}h_{i}(\mathbf {A} ),$

then,

$f(\mathbf {A} +\varepsilon \mathbf {X} )=f(\mathbf {A} )+\varepsilon \sum _{i}g_{i}(\mathbf {A} )\mathbf {X} h_{i}(\mathbf {A} )+{\mathcal {O}}\left(\varepsilon ^{2}\right).$

Given a positive integer$n$,

${\begin{aligned}{\frac {\mathrm {d} \mathbf {A} ^{n}}{\mathrm {d} t}}&=\sum _{i=1}^{n}\mathbf {A} ^{i-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{n-i},\\{\frac {\mathrm {d} \mathbf {A} ^{-n}}{\mathrm {d} t}}&=-\sum _{i=1}^{n}\mathbf {A} ^{-i}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-(n+1-i)}.\end{aligned}}$

Therefore,

${\begin{aligned}(\mathbf {A} +\varepsilon \mathbf {X} )^{n}&=\mathbf {A} ^{n}+\varepsilon \sum _{i=1}^{n}\mathbf {A} ^{i-1}\mathbf {X} \mathbf {A} ^{n-i}+{\mathcal {O}}\left(\varepsilon ^{2}\right),\\(\mathbf {A} +\varepsilon \mathbf {X} )^{-n}&=\mathbf {A} ^{-n}-\varepsilon \sum _{i=1}^{n}\mathbf {A} ^{-i}\mathbf {X} \mathbf {A} ^{-(n+1-i)}+{\mathcal {O}}\left(\varepsilon ^{2}\right).\end{aligned}}$

Generalized inverse

Some of the properties of inverse matrices are shared bygeneralized inverses(for example, theMoore–Penrose inverse), which can be defined for anym-by-nmatrix.

Applications

For most practical applications, it isnotnecessary to invert a matrix to solve asystem of linear equations;however, for a unique solution, itisnecessary that the matrix involved be invertible.

Decomposition techniques likeLU decompositionare much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.

Regression/least squares

Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.

Matrix inverses in real-time simulations

Matrix inversion plays a significant role incomputer graphics,particularly in3D graphicsrendering and 3D simulations. Examples include screen-to-worldray casting,world-to-subspace-to-world object transformations, and physical simulations.

Matrix inverses in MIMO wireless communication

Matrix inversion also plays a significant role in theMIMO(Multiple-Input, Multiple-Output) technology inwireless communications.The MIMO system consists ofNtransmit andMreceive antennas. Unique signals, occupying the samefrequency band,are sent viaNtransmit antennas and are received viaMreceive antennas. The signal arriving at each receive antenna will be alinear combinationof theNtransmitted signals forming anN×Mtransmission matrixH.It is crucial for the matrixHto be invertible for the receiver to be able to figure out the transmitted information.

See also

• Binomial inverse theorem
• LU decomposition
• Matrix decomposition
• Matrix square root
• Minor (linear algebra)
• Partial inverse of a matrix
• Pseudoinverse
• Rybicki Press algorithm
• Singular value decomposition
• Woodbury matrix identity

Further reading

• "Inversion of a matrix",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Cormen, Thomas H.;Leiserson, Charles E.;Rivest, Ronald L.;Stein, Clifford(2001) [1990]. "28.4: Inverting matrices".Introduction to Algorithms(2nd ed.). MIT Press and McGraw-Hill. pp. 755–760.ISBN0-262-03293-7.
• Bernstein, Dennis S. (2009).Matrix Mathematics: Theory, Facts, and Formulas(2nd ed.). Princeton University Press.ISBN978-0691140391– viaGoogle Books.
• Petersen, Kaare Brandt; Pedersen, Michael Syskind (November 15, 2012)."The Matrix Cookbook"(PDF).pp. 17–23.

External links

• Sanderson, Grant (August 15, 2016)."Inverse Matrices, Column Space and Null Space".Essence of Linear Algebra.Archivedfrom the original on 2021-11-03 – viaYouTube.
• Strang, Gilbert."Linear Algebra Lecture on Inverse Matrices".MIT OpenCourseWare.
• Moore-Penrose Inverse Matrix