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Computación y Sistemas

versión impresa ISSN 1405-5546

Comp. y Sist. vol.20 no.2 México abr./jun. 2016

http://dx.doi.org/10.13053/cys-20-2-2083 

Articles

Relationship between the Inverses of a Matrix and a Submatrix

E. Juárez-Ruiz1  * 

R. Cortés-Maldonado2 

F. Pérez-Rodríguez2 

1Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias de la Electrónica, Puebla, Mexico. estelajuarez@correo.buap.mx.

2Benemérita Universidad Autónoma de Puebla, Instituto de Física, Puebla, Mexico. rcortes@ifuap.buap.mx, fperez@ifuap.buap.mx.

Abstract

A simple and straightforward formula for computing the inverse of a submatrix in terms of the inverse of the original matrix is derived. General formulas for the inverse of submatrices of order n - k as well as block submatrices are derived. The number of additions (or subtractions) and multiplications (or divisions) on the formula is calculated. A variety of numerical results are shown.

Keywords: Matrix inverse; submatrix; discrete Fourier transform

1 Introduction

There are a number of situations in which the inverse of a matrix must be computed. For example, in statistics 17, where the inverse can provide important statistical information in certain matrix iterations arising in eigenvalue-related problems.

Direct methods for calculating the inverse of matrices include LU Decomposition, Cholesky Decomposition, and Gaussian Elimination 12, 17.

In Vandermonde matrices

which arise in many approximation and interpolation problems, V is non-singular if scalars 𝛼i , i = 0, ..., n are different. The inverse of V can be calculated explicitly with 6n2 flops (see 17, p. 416). El-Mikkawy 11 provides an explicit expression for the inverse of generalized Vandermonde matrices by using elementary symmetric functions. Fourier matrices obtained from the Discrete Fourier Transform (DFT) are Vandermonde matrices with known inverses 12, 17.

Let A be a non-singular matrix and A-1 be its inverse. Sometimes, it is necessary to determine the inverse of an invertible submatrix of A. This situation is common in applied physics for superconductivity computations 15, photonic crystals 8, 21, metal-dielectric materials 25, and bianisotropic metamaterials 22.

In general, computation of the inverse of a submatrix from a matrix with the known inverse is not direct. Quite recently, Chang 9 provided a recursive method for calculating the inverse of submatrices located at the upper left corner of A.

In this paper, we aim to calculate the inverse of a non-singular submatrix in terms of the elements of the inverse of the original matrix. We compare the number of operations in our method with those of the Sherman-Morrison method and the LU Decomposition.

This problem is directly related to how to calculate the inverse of a perturbed matrix (A + D)-1, where D is a perturbation matrix of A 10, 14, 19, 24. This matrix inverse has been calculated in various disciplines with different applications, derived from the Sherman-Morrison formula 5, 23:

(A- uvT)-1= A-1 +(A-1u)(vTA-1)1-vTA-1u, (1)

where u, 𝑣 ∈ ℝn are column vectors, from the Sherman-Morrison-Woodbury formula 14, 16:

[A-UV ]-1 = A-1 +A-1U(I-VA-1U)-1 VA-1,

or from its block-partitioned matrix form 14:

M-1 =(A-1 +A-1UC-1VA-1-C-1VA-1                   -A-1UC-1C-1), (2)

Where

M=AUVD. (3)

and C = D - VA -1U is the Schur complement of A.

Particularly, formula (2) has been applied by inverting a matrix with the enlargement method 13, which uses the same formula to express the inverse of a leading principal submatrix of order k in terms of a previously calculated submatrix of order (k - 1).

Applications of these formulas have been described in various papers. For example, Hager 14 discusses applications in statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations; Maponi 18 and Bru et al., 7 in solving linear systems of equations; Arsham, Grad, and Jaklic 4 in linear programming; Akgün, Garcelon, and Haftka 1 in structural reanalysis; and Alshehri 3 in the multi-period demand response management problem.

Now, we show a case where the perturbation matrix A - u𝑣T can be used to solve the problem of calculating the inverse of an invertible submatrix of order n - 1 of a known invertible matrix.

Let Ap-,q- be the submatrix obtained from A by eliminating the p - th row and q -th column. We state A - u𝑣T by defining u = Aq - ep, where Aq is the q - th column vector of A, ep ∈ ℝn is the p - th canonical column vector, and v = eq is the q - th canonical column vector. With these definitions, A - u𝑣T is equal to A except in its q - th column, which is equal to ep. By applying the Sherman-Morrison formula to calculate (A - u𝑣T)-1 , then Ap-,q--1 obtained by eliminating the q - th row and p - th column of (A - u𝑣T)-1.

The following example illustrates this procedure.

Let A and A-1 be

A= 1462-13325, A-1=127-11-818-1-139710-9

Let A2-,3-=1432, then

u=A3- e2=(6,3-1,5)T,          v=(0,0,1)T,

and

A-uvT=1402-11320

Since A - u𝑣T is invertible, by using the Sherman-Morrison formula we obtain

By eliminating the 3rd row and the 2nd column, we obtain

(A- uvT)-3;12=110(-243-1)=A2-;3--1,

which is the inverse of the submatrix.

If the number of additions and subtractions (NAS) and the number of multiplications and divisions (NMD) are considered separately, the Sherman-Morrison formula provides a method for calculating the inverse of a submatrix of order n -1, with

NAS = 2n(2n - 1); NMD = n(5n + 1), (4)

where n is the order of the original matrix. The result is obtained by doing a simple sum of each algebraic operation performed on the different steps of the algorithm.

In this paper, we show a simpler, more direct formula with

NAS = (n-1)(n-1);NMD = 2(n-i)(n-i). (5)

The paper is organized as follows. In the next section, we show a formula for calculating each element of the inverse of a non-singular submatrix of order n - 1 in terms of the elements of the inverse of the original matrix. An example of the use of the formula is illustrated in Section 3. The formula is implemented computationally in Section 4 on MatLab and Fortran 90 for a Fourier matrix, comparing the formula's runtime with respect to the already implemented algorithms in each programming language that are based on LU decomposition. Then, in Section 5, a general formula for the inverse of any square submatrix of a given n x n matrix is obtained. Finally, in Section 6, the relationship between the inverses of block submatrices and their original matrix, which was used in 8, 22,25, is derived.

2 Submatrices of Order n - 1

In the sequel, we consider the vector space Fnxn of matrices over the real or complex field.

Let AFnxn, , A = (aij), i,j = 1,...,n be invertible, and let A-1= (bij), i,j = 1,..., n be its inverse. Then, we obtain

bij = (-1)i+jdetAj-;i-detA. (6)

Let M=Ap-,q- be a submatrix of A. For our purposes, we will use the following notation:

M = (aij), i=1,,p-1,p + 1,,n,j=1,,q-1,q + 1,,n,

or, in short,

M = (aij), i,j = 1:n,ip,jq.

Note that Ap-,q- is invertible ⟺ 𝑏𝑞𝑝 ≠ 0..

Next, we derive the formula for the calculation of the inverse of M-1 = (mij).

Theorem 2.1. Let A = (aij) be a nonsingular matrix of order n, and let A-1 = (bij) be its inverse. If apq and bqp are both not null for certain p, q ∈ {1,..., n}, then the submatrix M=Ap-,q- is invertible, and its inverse M-1 = (mij) is a matrix of order (n - 1) defined as

mij= bij -bipbqjbqp,i,j = 1:n,iq,jp (7)

Proof. Since A-1 is the inverse of A and, reciprocally, A-1 A = AA-1 = ln where ln is the identity matrix of order n. Thus,

i,j = 1:n,k=1nbikakj=δij,i,j = 1:n,k=1naikbkj=δij,,

where δij the Kronecker's delta, being equal to 1 if i = j and to 0 if I ≠ j. These equations can be expressed as

(8)

(9)

We define D = (dij) ∈ F(n-1)x(n-1) as the matrix

dij= δij-bipapj,i,j = 1:n; iq,jq,

where p and q indicate the number of the row and the column, respectively, which are eliminated from matrix A to obtain the submatrix M=Ap-,q-.

Matrix D can be expressed as

D = In-1-uvT, (10)

where u = (b1p,...,b(q-1)p, b(q+1)p...,bnp)T is the p - th column of A-1 after eliminating its q - th component. Analogously, vector ν = (ap1,…,ap(q -1), ap(q +1)...., apn)T is the p - th row of matrix A after eliminating its q - th component.

The inverse of D in Eq. (10) can be calculated by using the Sherman-Morrison formula (1), which contains the scalar 1 - νTu, and by using Eq. (9) we can see that

Thus, if apqbpq≠0 (i.e., both aqp and bqp are nonzero), D is invertible and, according to Eq. (1), we obtain

On the other hand, D can be expressed as a matrix form by using Eq. (8) such that

NM = D, (11)

where N is the submatrix of A-1 defined as

(12)

According to Eq. (11), D-1NM=ln-1. Then,

Substituting u,νT and using matrix N in Eq. (12), the elements mij of matrix M-1 are given by

Finally, using Eq. (9) we obtain the formula

In this theorem, the condition apq0 is necessary due to the use of the Sherman-Morrison formula; however, this hypothesis is removed in the theorem below.

Theorem 2.2. Let A be an invertible matrix of order n, and let A-1 = (bij) be its inverse. If bqp0 for some q, p ∈ {1, ...,n}, then M=Ap-,q- is invertible and its inverse M-1 = (mij) is given by Eq. (7).

Proof. It is sufficient to prove that submatrices M and M-1 satisfy the relation M-1M= In-1 (see 6). Since M = (aij), i,j = 1:n, I ± p,jq and M-1 = (mij), i,j = 1: n , iq,jq, the elements of their product M-1M = (cij) are

Substituting mik in Eq. (7),

by Eq. (8)

since jq, we obtain δqj = 0. ∎

By doing a simple sum of the operations required to obtain the inverse of submatrix M=Ap-,q- in Eq. (7), NAS and NMD are confirmed to be as in Eq. (5).

3 Example

Consider the DFT 𝓕 of the sequence of n complex numbers x0, ..., xn-1 into the n complex numbers y0,… ,yn-1 according to the formula:

This linear transformation can be expressed in terms of the n x n Vandermonde matrix F as

y=Fx= Fx,

where y = (y0, ..., yn-1)T, x = (x0, ..., xn-1)T ∈ ℂn, and F is

(13)

The inverse of matrix F corresponds to the Inverse Discrete Fourier Transform

x=F-1y=F-1y

where F-1 is given by F-1 =1nF* (the asterisk denotes complex conjugate):

Now, let us apply Theorem 2.2 to calculate the inverses of submatrices of order n - 1 of the matrix F in Eq. (13). To achieve this purpose, it is convenient to express matrices F and F-1 in the form

(14)

Note that gqp ≠ 0, for all q, p ∈ {1, ...,n}, then any submatrix M=Fp-,q- of F is invertible by using Theorem 2.2, and its inverse M-1= (mkl) is given by (7) as

(15)

It should be emphasized that Eq. (15) provides the inverse of any submatrix of order n - 1 of matrix F in (13).

For the specific case n = 4, F has the form

F=111(-i)1      11(-i)2(-i)31(-i)21(-i)3      (-i)4(-i)6(-i)6(-i)9

or equivalently

111i      11-1i1-11i      1-1-1-i

And its inverse is given by

i. If M=F4-,2-, then by using formula (15), we directly obtain

ii. If M=F4-,4- is a principal submatrix then we obtain

4 Computational Implementation

First, we calculate the number of operations of the Sherman-Morrison method, formula in Eq. (7), and the LU algorithm. By using equations (4) and (5), the total number of operations to compute the matrix inverse with the Sherman-Morrison formula in Eq. (1) is 2n(2n - 1) + n(5n + 1) = 9n2 - n = 0(n2); with the formula in Eq. (7), 3(n - 1)2 = 0(n2); and with LU Decomposition, 0(n3) operations are required 2. In the specific case of Vandermonde matrices, we need 6n2 flops.

Although the number of operations with the Sherman-Morrison formula and the formula in Eq. (7) are of the same order, the slopes of the polynomial functions given by the number of operations of each method are 18 and 6, respectively, so we argue that the algorithm provided in this paper is more efficient. With the Vandermonde matrices, the slope of the function given by the number of operations is 12.

In the remaining part of this section, we compare the results of the implementation of formula (7) with LU MatLab algorithm on v.R2008a and Fortran 90 for the specific case of Vandermonde matrices of DFT (see Section 3). The algorithms were executed on a notebook with 2.27 GHz Intel Core i3 processor and a 4 GB RAM memory.

To implement the algorithm, row 4 and column 2 were eliminated in order to obtain the submatrix of order n - 1.

Figure 1 shows the results of comparing the matrix size with runtime on MatLab. For matrices of order 600 approximately, the algorithm performance in Equation (7) is similar to the performance of MatLab's LU algorithm. However, for higher orders, the traditional algorithm requires higher runtimes, whereas formula (7) maintains small values for matrices of order 3 x 103 approximately.

Fig. 1 Implementation of Equation (7) in comparison to the LU algorithm on MatLab 

In this case, the runtime is about 3 seconds in comparison to 90 seconds of the LU algorithm.

In Figure 2, the implementation results in Fortran 90 are presented. Note that the same pattern with the runtime variant increases significantly. Therefore, for a matrix of order 3 x 103 approximately, the LU algorithm runtime is about 1300 seconds.

Fig. 2 Implementation of Eq. (7) in comparison to the LU algorithm on Fortran 90 

Finally, in Figure 3, the performance of Equation (7) in both computational programs is exposed. Note that there is no significant difference on runtime performance, obtaining values of the same order of magnitude. For matrices of order 3 x 103 approximately, the runtime does not exceed three seconds. This is an indicator that algorithm performance does not depend on software.

Fig. 3 Computational comparison between MatLab and Fortran 90 programs 

5 Submatrices of Order n - k

5.1. Iterative Procedure

The derived relation (7) between the inverse of a submatrix Ap-,q- of order n - 1 with the inverse A-1 = (bij) of the original matrix A can be iteratively applied to calculate the inverse of a submatrix of order (n - k), 1 ≤ k < n.

Let Mk=Ap1-,, pk-; q1-, qk- be a submatrix of order (n - k) obtained from a matrix A of order n by eliminating its p1-,..., pk - th rows and its q1-,..., qk - th columns. Then, the inverse Mk-1 = (mij(k)) of the submatrix Mk can be obtained by applying the iterative procedure:

(16)

This algorithm is applicable by using Theorem 2.2 if

(17)

i.e., all submatrices Ml,(I = 1: k) are invertible.

5.2 General Formula

Let us apply the iterative procedure described above to obtain explicit expressions for the elements mij(l) of the inverses of square submatrices in terms of determinants containing the elements bij of A-1 .

Case M1. We can express formula (7) for mij(1) of matrix M1-1 in (16) as follows:

(18)

In particular mq2p2(1), is given by

(19)

Case M2. Consider the invertible submatrix M1=Ap1-; q1- (i.e., bq1,p1 0), and let M1-1=mij(1) be its inverse. Let p2, q2 ∈ {1, ...,n} such that p1p2 , q1q2. If the element mq2p2(1) (17) of the matrix M1-1 is not null mq2p2(1) 0, then the submatrix M2=Ap1-, p2-; q1-,q2- of order (n - 2), obtained from M1=Ap1-; q1- by eliminating its p2-th row and q2-th column, is invertible. By (16) and (18), the elements mij(2) of matrix M2-1 can be expressed as

After simplifying, we obtain

Thus,

In this case, therefore, we have the following theorem.

Theorem 5.1. Let A be a nonsingular matrix of order n3, and let A-1 = (bij) be its inverse. If the submatrix of order 2

bq1p1bq1p2bq2p1bq2p2 (20)

of A-1 has non-null leading principal minors, for certain p1, p2, q1, q2 ∈ {1,2,...,n} with p1p2, q1q2, then M2=Ap1-, p2-; q1-,q2- is invertible and its M2-1= mij(2)is given by

(21)

Proof. The leading principal minors of submatrix (20) are:

bq1p1,bq1p1bq1p2bq2p1bq2p2

If these minors are different from zero, then mq2p2(1) in (19) is not null. Subsequently, if conditions in (17) bq1,p1 0,  mq2p2(1)0, are fully satisfied, then M2 is invertible. The elements of M2-1 can be calculated by using formulas in (16), which can be expressed again in the form of (21). ∎

Case Mk. The above results obtained for cases M1 and M2 allow us to infer a general formula for Mk, with 1k < n.

Theorem 5.2. Let A be a nonsingular matrix of order n, and let A-1 = (bij) be its inverse. Let kN such that k < n. If the submatrix of order k x k

bq1p1bq1p2bq2p1bq2p2    bq2p2bq2p2bqkp1bqkp2    bqkpk (22)

of A-1 has non-null leading principal minors for certain p1, …, pk, q1, …, qk ∈ {1,...,n} satisfying pj1pj2 for j1j2 and qi1qi2 for i1i2 , then the submatrix Mk=Ap1-, , pk-; q1-,,qk-of A is invertible and its inverse Mk-1= mij(k) is a matrix of order (n - k) with elements defined by

(23)

Proof. Let us demonstrate the theorem by mathematical induction.

Step 1. Let us verify that the proposition of the theorem is true for case M1 . If the 1 x 1 submatrix

bq1p1

of A-1 has non-null leading principal minors, i.e bq1p1 0, then the submatrix M1=Ap1-; q1- is invertible and its inverse M1-1= mij(1) is given by formula (7) from Theorem 2.2. The general expression (23) is another form of Eq. (7) as shown in Eq. (18).

Step 2. Let us suppose that the proposition is true for case Mk -1. Thus, if the submatrix of A-1 of order k - 1

bq1p1bqkp1bqk-1p1bqk-1pk-1

has non-null leading principal minors, the submatrix Mk-1=Ap1-, , pk-1-; q1-,,qk-1- of A is invertible and its inverse Mk-1-1= mij(k-1) is the matrix of order (n - k + 1) given by

(24)

If the conditions in (17) are satisfied, the elements mij(k) of matrix Mk-1 are expressed in terms of the elements mij(k-1) of Mk-1-1 according to Eq. (16). Such conditions demand that leading principal minors of matrix (22) be non-null. In fact, note that the elements bq1p1,mq2p2(1),, mqkpkk-1, appearing in the denominators of Eq. (16), turn out to be proportional to those minors, see Eq. (24). In the sequel, we denote the elements mij(k) (16) as

(25)

where we have used the following notation:

Using Eq. (2) for the determinant of a block-partitioned matrix (3), we directly obtain

(26)

This result agrees with formula (23). In fact, by expressing (23) as

and using Eq. (2) for the determinant of a block-partitioned matrix (3), we directly obtain

(26)

This result agrees with formula (23). In fact, by expressing (23) as

and using Eq. (5), we obtain

Subsequently, this formula is reduced to the expression

which evidently agrees with (26). It implies that this proposition is true for all k values. ∎

Note that in the specific case of k = n - 1 in Theorem 5.2, the submatrix Mn-1=Ap1-, , pn-1-; q1-,,qn-1- of A is a 1 x 1 matrix and its inverse is Mn-1-1= mij(n-1), where

(27)

Then, indexes i and j, respectively, take the remaining value from the integers in {1,...,n}. Permutating the rows and columns of the determinant in the numerator of expression (27), we obtain

By using (6) to calculate the elements of the matrix inverse of A -1((A -1)-1 = A), we obtain the expected result

Mij(n-1)= 1aij

6 Block Submatrices

We generalize the relationship between the inverses of a matrix and their submatrices, which is derived in Section 2, to the case of block-partitioned matrices having square blocks of the same size.

Theorem 6.1. Let A = (Aij) be a nonsingular block matrix of order ns, and let A-1 = (Bij) be its inverse, where Bij is a s x s square block matrix, (1 ≤ i; jn) . If Bqp is invertible for certain q, p ∈ {1, ...,n}, then the block-partitioned submatrix M = Ap-q- obtained by eliminating the p- th block row and the q- th block column of A is invertible, and its inverse M-1 = (Mij) of order (n -1)s is given by

Mij=Bij-BipBqp-1Bqj,i,j=1:n,i q,jp. (28)

Proof. The demonstration follows the same procedure as Theorem 2.2.

7 Conclusions

In summary, we have obtained a formula (Eq. (7)) that allows us to calculate the inverse of a submatrix of order (n - 1) in terms of the inverse A-1 of the original n x n matrix A. By applying such a formula iteratively, we have been able to derive an explicit relationship (23) between the inverse of an arbitrary square submatrix and its inverse A-1.

In addition, we have tested the computational efficiency of the formula's runtime when compared with the LU Decomposition for the case of Fourier matrices. We have also generalized formula (5) for the case of inverses of block-partitioned matrices with square blocks of the same size s, see Eq. (28). The relationship in Eq. (28) is particularly useful when the known inverse of the matrix is a very large order (ns » 1), and it is necessary to calculate the inverse of a submatrix of order (n - 1)s.

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Received: November 21, 2014; Accepted: May 20, 2016

*Corresponding author is E. Juárez Ruiz.

E. Juárez Ruiz has a Ph.D. degree in Mathematics from Benemérita Universidad Autónoma de Puebla. She works as a Research Professor at the Faculty of Electronics in the same university. Her research interests include applied mathematics and educational models mediated by Information and Communications Technologies (ICT).

R. Cortés Maldonado has a B.Sc. degree in Electronics and an M.Sc. degree in Physics from Benemérita Universidad Autónoma de Puebla. Currently, he works on his Ph.D. research project on magnetic properties of superconductors.

F. Pérez Rodríguez has a Ph.D. degree in Physics and Mathematics from Kharkov State University. He is a National Investigator, Level 3. He is interested in optical and acoustical properties of solids, in metamaterials, and in superconductivity.

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