Full-rank block LDL∗ decomposition and the inverses of n×n block matrices

JAMCJ Appl Math Comput (2012) 40:569–586DOI 10.1007/s12190-012-0579-3

C O M P U TAT I O NA L M AT H E M AT I C S

Full-rank block LDL∗ decomposition and the inversesof n × n block matrices

Ivan Stanimirovic

Received: 20 March 2012 / Published online: 22 June 2012© Korean Society for Computational and Applied Mathematics 2012

Abstract Full-rank block LDL∗ decomposition of a Hermitian n × n block matrix A

is examined, where the iterative procedure evaluating the sub-matrices appearing inL and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n×n block matrix. The method for the calculation ofthe Moore-Penrose inverse of an arbitrary 2 × 2 block matrix is also provided. There-fore, matrix products A∗A and AA∗ and the corresponding full-rank block LDL∗factorizations are observed. Also, a simple explicit formulae calculating the solutionvector components of the normal system of equations is stated, where the LDL∗ de-composition of the system matrix is done.

Keywords Full-rank · LDL∗ decomposition · Solution vector · Inverse matrix ·Hermitian matrix · Block matrix · Moore-Penrose inverse matrix

Mathematics Subject Classification 15A09 · 15A23

1 Introduction and preliminary results

The Cholesky decomposition is often used to calculate the inverse matrix A−1 andthe determinant of A. For a given Hermitian positive definite matrix A (see [1]), thereexists a nonsingular lower triangular matrix L such that

A = LL∗.

If A is not a positive definite matrix, the matrix L can be singular as well. One canalso impose that the diagonal elements of the matrix L are all positive, in which case

I. Stanimirovic (�)Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbiae-mail: [email protected]

mailto:[email protected]

570 I. Stanimirovic

the corresponding factorization is unique. The difficulty involving the Cholesky de-composition is the appearance of square root entries in matrix L, which is sometimescalled the “square root” of A.

The procedure for block Cholesky decomposition involves computation of thesquare root of a diagonal block and then multiplication of the inverse of a triangularblock by a square block. Finding the square root of a block matrix can be difficult andtime consuming job. It can be done by using the same equations, where each entrybecomes block matrix. Some attempts have been made to find the approximation ofblock LU and block Cholesky factorization (see [2]). Also, multiplying the inverse ofa triangular block by a block can be accomplished by the back substitution process.

Exchanging the LL∗ decomposition by LDL∗ in order to avoid generation of squareroot entries is a well explained technique from linear algebra (see for instance Goluband Van Loan in [5]). LDL∗ factorization requires less the computation of Gaussianelimination (the same as LU decomposition), and it is stable. But, its main advantageis that it avoids computing the square root entries, making it more efficient and sim-ple than Cholesky factorization. The LDL∗ bypasses the production of square rootentries. Instead, one additional diagonal matrix D is generated, but the total evalua-tion is the same to the Cholesky decomposition. Notice that for the given Hermitianmatrix A, the matrix D must have positive entries.

Consider an arbitrary set of normal matrix equations. It can be expressed in thefollowing matrix notation:

Ax = B,

where x is an n-dimensional solutions vector. Here we provide some preliminaryresults for the evaluation of the solution vector x, based on the LDL∗ factorization ofthe matrix A. Suppose that the matrix A of the given normal equations is Hermitianand positive definite. Then there exist a nonsingular lower triangular matrix L and anonsingular diagonal matrix D such that

A = LDL∗.

The inverse matrix A−1 and the solution vector x can be found as follows:

A−1 = L∗−1 · D−1 · L−1,

x = L∗−1 · D−1 · L−1 · B.(1.1)

Notice that these equalities involve the inverse of a triangular and a diagonal matrix.However, this is not required, because one can divide a matrix by a triangular matrixwithout computing the inverse. Therefore, consider the following shift:

C = DL∗x.

Then L · C = B and therefore the following expressions are valid:

ci = bi − ∑i−1k=1 li,k · ck

li,i, for i = 1, n. (1.2)

Full-rank block LDL∗ decomposition 571

Observe that the matrix product DL∗ is the upper triangular matrix, and therefore thefollowing equations hold:

xn = cn

dn,n

, (1.3)

xi = ci − ∑nk=i+1 di,i lk,i · xk

di,i

, for i = n − 1,1. (1.4)

The inverses of 2 × 2 block matrices have been well studied (see [7, 8]), and oftenappear in many subjects. Some explicit inverse formulae for a 2 × 2 block matrixwere given in [7, 11], which are based on the Schur complement D − CA−1B of A.

Lemma 1.1 Consider a 2 × 2 nonsingular partitioned matrix

M =[

A B

C D

]

,

where A ∈ Ck×k , B ∈ Ck×l , C ∈ Cl×k and D ∈ Cl×l . If the sub-matrix A is nonsin-gular, then the Schur complement SA = D − CA−1B of A in M is also nonsingular,and the inverse of M has the following form

M−1 =[A−1 + A−1BS−1

A CA−1 −A−1BS−1A

−S−1A CA−1 S−1

A

]

. (1.5)

Many papers provide methods for the evaluation of the Moore-Penrose inverse of a2 × 2 block matrices (see for example [6]). Also, some representations for the Drazininverse of a 2 × 2 block matrices have been developed in [4]. However, the generalcase of n × n partitioned matrix, n > 2, has not been explained as much (the Moore-Penrose inverses of m × n block matrices were examined in [12]). Therefore, we tryto develop methods involving n × n block matrices and then observe the particularcase of 2 × 2 block matrices.

The paper has three objectives. Our main one is to develop a procedure for theblock LDL∗ decomposition, for the general case of n × n block Hermitian matrixA ∈ Cm×m. In this way difficulties with evaluation of square roots of sub-matrices(often called blocks), will be avoided. Also, the extensions considering symboliccomputation of polynomial block matrices (and matrices with rational entries [10]),which occur from the square roots appearance, will be feasible.

Then, two objectives are to take the advantages of square-root-free decompositionand develop methods providing the inverse and the Moore-Penrose inverse of a blockmatrix A. We continue the idea from [9], where the LDL∗ factorization is used forthe computation of generalized inverses. For the sake of completeness we provide thedefinition of the Moore-Penrose inverse, as stated in [3].

Definition 1.1 For an arbitrary matrix A ∈ Cm×n we observe the following matrixequations of the unknown X, where ∗ denotes conjugate transpose:

(1) AXA = A (2) XAX = X (3) (AX)∗ = AX (4) (XA)∗ = XA.

The matrix X = A† is the Moore-Penrose inverse of the matrix A if it satisfies (1)–(4).

572 I. Stanimirovic

The paper is organized as follows. In the second section the full-rank block LDL∗decomposition of a n × n block matrix is observed, where the procedure calculatingthe sub-matrices of L and D is established. Inverses of Hermitian n × n block matri-ces and the particular case of 2 × 2 block matrices are examined in the third section.Thereat, the methods for the evaluation of these inverses are based on the block LDL∗decomposition. In the fourth section we develop a method for the calculation of theMoore-Penrose inverse of a 2 × 2 block matrix.

2 Full-rank block LDL∗ decomposition of n × n block matrices

Let us suppose that a Hermitian matrix A is partitioned to n × n blocks, such thatA = [Aij ], 1 ≤ i, j ≤ n, and the diagonal blocks are square matrices. A goal is to de-termine a lower block triangular matrix L and a block diagonal matrix D, such thatthe equation A = LDL∗ holds and the dimensions of the sub-matrices Li,j and Di,j

are di × dj . A simple modification of the iterative procedure for the LDL∗ decompo-sition which holds for the block entries of D and L can be stated.

Now we provide the statement giving the effective procedure calculating the sub-matrices from the full-rank matrices L and D.

Theorem 2.1 Consider a Hermitian block matrix A = [Aij ]n×n of the rank r , parti-tioned in such manner that the dimensions of each sub-matrix Ai,j are di × dj , andthere exists a positive integer m ≤ n such that the expression

∑mk=1 dk = r holds. Let

L = [Lij ]n×m and D = [Dij ]m×m be block matrices, where the dimensions of eachsub-matrix Lij and Dij are di × dj . If the following matrix equations hold for eachj = 1,m:

Djj = Ajj −j−1∑

k=1

Ljk · Dkk · L∗jk, (2.6)

Lij =(

Aij −j−1∑

k=1

Lik · Dkk · L∗jk

)

· D−1jj , i = j + 1, n, (2.7)

where the sub-matrices Djj , j = 1,m are non-singular, then LDL∗ is the full-rankdecomposition of the matrix A.

Proof Since A = LDL∗, notice that for an arbitrary indices 1 ≤ i, j ≤ n the followingis valid:

Aij =m∑

k=1

LikDkkL∗jk =

min{i,j}∑

k=1

LikDkkL∗jk,

since the next matrix equalities are satisfied for i = 1,m:


Lii = Idi,

Lij = 0di×dj, Dij = 0di×dj

, Dji = 0dj ×di, j = i + 1,m.

Therefore, for each index j = 1,m

Ajj =j∑

k=1

LjkDkkL∗jk = Djj +

j−1∑

k=1

LjkDkkL∗jk

is a valid equation, from which (2.6) holds.Next, for the case of i > j we have

Aij =j∑

k=1

LikDkkL∗jk = LijDjjL

∗jj +

j−1∑

k=1

LikDkkL∗jk.

Therefore, the following is satisfied:

LijDjj = Aij −j−1∑

k=1

LikDkkL∗jk,

and (2.7) is validated, since the sub-matrix Djj is non-singular.Since the expression

∑mk=1 dk = r holds, the matrices L = [Lij ]n×m and D =

[Dij ]m×m are of the rank r , according to (2.6) and (2.7). Therefore, LDL∗ is thefull-rank decomposition of the matrix A. �

Obviously, the iterative procedure for the block LDL∗ decomposition of a Hermi-tian matrix is attained from (2.6) and (2.7). In every iteration, sub-matrices Djj andLij , i > j can be evaluated from the previous results and the known blocks Aij .

Notice that multiplying a block matrix by the inverse of a block can be accom-plished by using the back substitution process. Also, the condition that the sub-matrices Djj , j = 1, n are non-singular can be too strong in some cases. These con-ditions can be avoided by using the Moore-Penrose inverses of Djj , in which case(2.7) becomes:

Lij =(

Aij −j−1∑

k=1

Lik · Dkk · L∗jk

)

· D†jj , i = j + 1, n. (2.8)

Remark 2.1 The iterative procedure for the LDL∗ decomposition of a block solutionvector is similar to the unit approach. Again the inverse of a triangular block multi-plied by a block can be accomplished by back substitution.

Therefore, as the sequel of Theorem 2.1, the following algorithm can be stated,generating the full-rank block matrices L, D of the dimensions smaller or equal tothe dimensions of the matrix A.

574 I. Stanimirovic

Algorithm 2.1 Full-rank block LDL∗ decomposition of a block matrix A

Require: Block matrix A = [Aij ]ni,j=1 of the rank r , where the dimensionsof a block Ai,j are di × dj , and for some m ∈ {1, . . . , n} the expression∑m

k=1 dk = r holds.1: Initialization: for i = 1,m do:

1.1: Set Lii = Idi.

1.2: For j = i + 1, n set Lij = 0di×dj.

1.2: For j = i + 1,m set Dij = 0di×dj, Dji = 0dj ×di

.

2: Evaluation: for j = 1,m do:2.1: Set Djj = Ajj − ∑j−1

k=1 Ljk · Dkk · L∗jk .

2.2: For i = j + 1, n set Lij = (Aij − ∑j−1k=1 Lik · Dkk · L∗

jk) · D−1jj .

Example 2.1 Consider the following 3 × 3 block matrix A partitioned as follows:

A =

⎡

⎢⎢⎢⎢⎢⎢⎣

5 6 7 5 6 76 9 10 −6 9 107 10 13 7 10 135 −6 7 5 −6 −76 9 10 −6 −9 127 10 13 −7 12 −13

⎤

⎥⎥⎥⎥⎥⎥⎦

.

After the initialization of the sub-matrices D12, D13, D21, D23, D31, D32, L12, L13,L23, perform Step 2 of Algorithm 2.1. For j = 1 the sub-matrix D11 is equal to A11,and therefore, according to Step 2.2:

L21 = A21 · D−111 =

[7 −12 60 1 0

]

,

L31 = A31 · D−111 = [

0 0 1].

In the second iteration, for j = 2 the sub-matrices D22 and L32 are evaluated:

D22 = A22 − L21D11L∗21 =

[−144 00 −18

]

,

L32 = (A32 − L31 · D11 · L∗

21

) · D−122 = [

772

119 u

].

Finally, for the case j = 3, we have that

D33 = A33 − L31 · D11 · L∗31 − L32 · D22 · L∗

32 =[

9

4

]

,


and therefore, A = LDL∗ is the full-rank block decomposition of A, where

L =

⎡

⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 07 −12 6 1 0 00 1 0 0 1 0

0 0 1 772

119 1

⎤

⎥⎥⎥⎥⎥⎥⎦

, D =

⎡

⎢⎢⎢⎢⎢⎢⎣

5 6 7 0 0 06 9 10 0 0 07 10 13 0 0 00 0 0 −144 0 00 0 0 0 −18 0

0 0 0 0 0 94

⎤

⎥⎥⎥⎥⎥⎥⎦

.

Notice that the total number of non-zero entries in matrices L and D is decreasingwith the further partitioning of the matrix A. Matrices L and D can be treated assparse at some point of partitioning.

Example 2.2 The main difference between the full-rank block LDL∗ decompositionand the standard one can be seen on the class of rank-deficient matrices. Therefore,let us consider the matrix A2 of the rank r = 2 from [13], partitioned in the followingway:

A2 =

⎡

⎢⎢⎢⎢⎢⎢⎣

a + 10 a + 9 a + 8 a + 7 a + 6 a + 5a + 9 a + 8 a + 7 a + 6 a + 5 a + 4a + 8 a + 7 a + 6 a + 5 a + 4 a + 3a + 7 a + 6 a + 5 a + 4 a + 3 a + 2a + 6 a + 5 a + 4 a + 3 a + 2 a + 1a + 5 a + 4 a + 3 a + 2 a + 1 a

⎤

⎥⎥⎥⎥⎥⎥⎦

.

By applying Algorithm 2.1 the following full-rank matrices are generated as the re-sult, for m = 1:

L =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 00 1

−1 2−2 3

−3 4−4 5

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

, D =[

10 + a 9 + a

9 + a 8 + a

]

.

Example 2.3 Consider the rank-deficient symbolic matrix F6 derived in [13], parti-tioned in the next manner:

F6 =

⎡

⎢⎢⎢⎢⎢⎢⎣

a + 6 a + 5 a + 4 a + 3 a + 2 a + 1a + 5 a + 5 a + 4 a + 3 a + 2 a + 1a + 4 a + 4 a + 4 a + 3 a + 2 a + 1a + 3 a + 3 a + 3 a + 3 a + 2 a + 1a + 2 a + 2 a + 2 a + 2 a + 1 a

a + 1 a + 1 a + 1 a + 1 a a − 1

⎤

⎥⎥⎥⎥⎥⎥⎦

.

Observe that rank(F6) = 5, and therefore the full-rank block LDL∗ factorization im-plies matrices L ∈ C(a)6×5

5 and D ∈ C(a)5×55 . By applying Algorithm 2.1 on the

576 I. Stanimirovic

matrix F6, for the case of n = 3, m = 2, the following rational matrices are gener-ated as the result:

L =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 00 1 0 0 0

0 4+a5+a

1 0 0

0 3+a5+a

0 1 0

0 2+a5+a

0 0 1

0 1+a5+a

0 −1 2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

, D =

⎡

⎢⎢⎢⎢⎢⎢⎣

a + 6 a + 5 0 0 0a + 5 a + 5 0 0 0

0 0 4+a5+a

3+a5+a

2+a5+a

0 0 3+a5+a

2(3+a)5+a

2(2+a)5+a

0 0 2+a5+a

2(2+a)5+a

1+2a5+a

⎤

⎥⎥⎥⎥⎥⎥⎦

.

3 Inverses of n × n block matrix

The matrices L and D, involved in block LDL∗ decomposition, are lower block tri-angular and block diagonal matrix, respectfully. Let us first examine the inverses ofthese types of block matrices.

Lemma 3.1 Consider a block diagonal matrix D with n sub-matrices on its diago-nal. Then D is invertible if and only if Di is square and non-singular for 1 ≤ i ≤ n,in which case D−1 is also block diagonal matrix of the form

D−1 =

⎡

⎢⎢⎢⎣

D−11 0 · · · 00 D−1

2 · · · 0...

.... . .

...

0 0 · · · D−1n

⎤

⎥⎥⎥⎦

.

Lemma 3.2 The inverse of a non-singular lower block triangular matrix

L =

⎡

⎢⎢⎢⎣

I 0 · · · 0L21 I · · · 0...

.... . .

...

Ln1 Ln2 · · · I

⎤

⎥⎥⎥⎦

,

is the lower block triangular matrix of the form

L−1 =

⎡

⎢⎢⎢⎣

I 0 · · · 0(L−1)21 I · · · 0

......

. . ....

(L−1)n1 (L−1)n2 · · · I

⎤

⎥⎥⎥⎦

,

where (L−1)i+1,i = −Li+1,i for each index i = 1, n − 1.


Proof Obviously, the matrix L−1 is the lower block diagonal matrix with identitymatrices on the diagonal.

From the equation LL−1 = I , we have the system of n2 matrix equations. For anarbitrary 1 ≤ i < n, multiplying the (i + 1)-th row of L by the i-th column of theinverse L−1 we have

Li+1,iI + I(L−1)

i+1,i= 0,

and therefore (L−1)i+1,i = −Li+1,i . �

As we previously determined, LDL∗ decomposition of a matrix A can be used todetermine its inverse as

A−1 = (L−1)∗ · D−1 · L−1. (3.9)

The method for the calculation of the Hermitian matrix inverse involving the blockLDL∗ decomposition is based on this equation. Therefore, we propose the followingtheorem.

Theorem 3.1 Consider the block LDL∗ decomposition of a non-singular Hermitianblock matrix A = [Aij ]ni,j=1, where the matrices L and D have the same partitioning

as A. If the matrix L and the sub-matrices Djj , j = 1, n are non-singular, then thefollowing matrix equations hold:

(A−1)

ij=

n∑

k=j

(L−1)∗

kiD−1

kk

(L−1)

kj, 1 ≤ i ≤ j ≤ n. (3.10)

Proof Follows from Lemma 3.1 and (3.9) by observing the partitioning of L−1 thesame as the partitioning of A. �

Corollary 3.1 Let A ∈ Cm×m be Hermitian 2 × 2 block matrix, partitioned in thefollowing way:

A =[A11 A12A∗

12 A22

]

.

Consider the block LDL∗ decomposition of the matrix A, where the matrices L andD have the same partitioning as A. If the sub-matrices D11, D22 are non-singular,then the inverse matrix A−1 has the following form:

A−1 =[

D−111 + L∗

21D−122 L21 −L∗

21D−122

−D−122 L21 D−1

22

]

. (3.11)

Proof The proof follows from Theorem 3.1 and Lemma 3.2, since the following iden-tities are valid:

A−1 =[

L∗11D

−111 L11 + L∗

21D−122 L21 −L∗

21D−122 L22

−L∗22D

−122 L21 L∗

22D−122 L22

]

578 I. Stanimirovic

=[

D−111 + L∗

21D−122 L21 −L∗

21D−122

−D−122 L21 D−1

22

]

. (3.12)

�

Notice that this representation of the inverse of 2 × 2 block matrix is simplerthan the similar one developed in [7], which is based on the Schur complement. Infact, the representation (3.11) is the sequel of Theorem 2.1 from [7], for the case ofD11 = A11, L21 = A∗

12D−111 and the Schur complement D22 = A22 − A∗

12D−111 A12.

Also, the conditions that D11, D22 are non-singular matrices are equivalent to theconditions of A11 and the Schur complement A22 − A∗

12A−111 A12 be non-singular.

Example 3.1 Denote the following partitioned matrix:

A =

⎡

⎢⎢⎢⎢⎣

−1 6 5 6 76 9 −6 0 105 −6 5 −6 −76 0 −6 −9 127 10 −7 12 −13

⎤

⎥⎥⎥⎥⎦

.

Then the matrices L and D from the block LDL∗ factorization of the matrix A are:

L =

⎡

⎢⎢⎢⎢⎢⎣

1 0 0 0 00 1 0 0 0

− 95

815 1 0 0

− 65

45 0 1 0

− 115

5245 0 0 1

⎤

⎥⎥⎥⎥⎥⎦

, D =

⎡

⎢⎢⎢⎢⎢⎣

−1 6 0 0 06 9 0 0 0

0 0 865

245

415

0 0 245 − 9

5625

0 0 415

625 − 1084

45

⎤

⎥⎥⎥⎥⎥⎦

.

In order to evaluate A−1, we have the following sequence of expressions:

(A−1)

11 =[−1 6

6 9

]−1

+⎡

⎢⎣

− 95

815

− 65

45

− 115

5245

⎤

⎥⎦

T

·⎡

⎢⎣

865

245

415

245 − 9

5625

415

625 − 1084

45

⎤

⎥⎦

−1

·⎡

⎢⎣

− 95

815

− 65

45

− 115

5245

⎤

⎥⎦

=[

99656 − 33

164

− 33164

96205

]

,

(A−1)

12 = (A−1)∗

21 = −⎡

⎢⎣

− 95

815

− 65

45

− 115

5245

⎤

⎥⎦

T

·⎡

⎢⎣

865

245

415

245 − 9

5625

415

625 − 1084

45

⎤

⎥⎦

−1

=[

13328

223984

75656

33410 − 481

1230 − 25164

]

,


(A−1)

22 =⎡

⎢⎣

865

245

415

245 − 9

5625

415

625 − 1084

45

⎤

⎥⎦

−1

=⎡

⎢⎣

69820 − 223

2460 − 15328

− 2232460

259820

53328

− 15328

53328

27656

⎤

⎥⎦ ,

and therefore the inverse matrix of A is equal to

A−1 =

⎡

⎢⎢⎢⎢⎢⎢⎣

99656 − 33

16413328

223984

75656

− 33164

96205

33410 − 481

1230 − 25164

13328

33410

69820 − 223

2460 − 15328

223984 − 481

1230 − 2232460

259820

53328

75656 − 25

164 − 15328

53328

27656

⎤

⎥⎥⎥⎥⎥⎥⎦

.

4 Moore-Penrose inverse of a 2 × 2 block matrix

The analogous statement to Theorem 3.1 can be stated, in order to compute theMoore-Penrose inverse of a Hermitian n × n block matrix, by observing its full-rankblock LDL∗ decomposition.

Theorem 4.1 Consider a Hermitian block matrix A = [Ai,j ]n×n of the rank r , wherethe dimensions of Ai,j are di × dj , and there exists a positive integer m ≤ n such thatthe expression

∑mk=1 dk = r holds. Let LDL∗ be the full-rank block decomposition

of the matrix A, where the sub-matrices Djj , j = 1,m are non-singular. Then thefollowing matrix equations hold:

(A†)

ij=

m∑

k=j

(L†)∗

kiD−1

kk

(L†)

kj, 1 ≤ i ≤ j ≤ n. (4.13)

Several methods for evaluation of the generalized inverses of a rational matrixwere introduced in [9]. Based on the LDL∗ decomposition of a corresponding matrixproducts, they provide an efficient way of calculating generalized inverses of constantmatrices. The following theorem from [9] gives the practical expression to computeMoore-Penrose inverse of a rational matrix.

Theorem 4.2 ([9]) Consider the rational matrix A ∈ C(x)m×nr . If LDL∗ is the full-

rank decomposition of a matrix (A∗A)2, where L ∈ C(x)m×r and D ∈ C(x)r×r , thenit is satisfied:

A† = L(L∗LDL∗L

)−1L∗(A∗A

)∗A∗. (4.14)

If LDL∗ is the full-rank decomposition of the matrix (AA∗)2, L ∈ C(x)m×r and D ∈C(x)r×r , then it is satisfied:

A† = A∗(AA∗)∗L

(L∗LDL∗L

)−1L∗. (4.15)

580 I. Stanimirovic

Now we are able to develop a method for the evaluation of the Moore-Penroseinverse of a 2 × 2 block matrix, given in the following corollary of Theorem 4.1 andLemma 1.1.

Theorem 4.3 Let A ∈ Cm×n and (A∗A)2 ∈ Cn×n be 2 × 2 block matrices of theforms

A =[A11 A12A21 A22

]

,(A∗A

)2 =[

B C

C∗ E

]

,

with the same partitioning. Let us use the following notations, in the case of thenonsingular matrix B:

X11 = B + N∗ + N + M∗EM,

X12 = C + ME,

X22 = E,

(4.16)

where the matrices M , N are determined by

M = C∗B−1,

N = CM.(4.17)

If the matrix X = [ X11 X12X∗

12 X22

]and the sub-matrices X11,B are nonsingular, then the

Moore-Penrose inverse of the matrix A is partitioned as follows:

A† =[

Y11Σ11 + (Y11M∗ + Y12)Σ21

(MY11 + Y21)(Σ11 + M∗Σ21) + (MY12 + Y22)Σ21

Y11Σ12 + (Y11M∗ + Y12)Σ22

(MY11 + Y21)(Σ12 + M∗Σ22) + (MY12 + Y22)Σ22

]

, (4.18)

where

Y11 = X−111 + X−1

11 X12Y22X∗12X

−111 ,

Y12 = −X−111 X12Y22,

Y21 = −Y22X∗12X

−111 ,

Y22 = (X22 − X∗

12X−111 X12

)−1,

Σ11 = (A∗

11A11 + A∗21A21

)A∗

11 + (A∗

11A12 + A∗21A22

)A∗

12,

Σ12 = (A∗

11A11 + A∗21A21

)A∗

21 + (A∗

11A12 + A∗21A22

)A∗

22,

Σ21 = (A∗

12A11 + A∗22A21

)A∗

11 + (A∗

12A12 + A∗22A22

)A∗

12,

Σ22 = (A∗

12A11 + A∗22A21

)A∗

21 + (A∗

12A12 + A∗22A22

)A∗

22.

Proof We shall consider the full-rank block LDL∗ decomposition of the Hermitianmatrix (A∗A)2, and make the appropriate unifications of some sub-matrices from L


and D if necessary, such that L and D are 2×2 block matrices. By observing the ma-trix product L∗LDL∗L, with the same partitioning, the following matrix equationshold:

(L∗LDL∗L

)11 = L∗

11L11D11L∗11L11

+ L∗21L21D11L

∗11L11 + L∗

11L11D11L∗21L21

+ L∗21L21D11L

∗21L21 + L∗

21L22D22L∗22L21

= D11 + L∗21L21D11 + D11L

∗21L21

+ L∗21L21D11L

∗21L21 + L∗

21D22L21

= B + B−1CC∗B−1B + BB−1CC∗B−1

+ B−1CC∗B−1BB−1CC∗B−1

+ B−1C(E − C∗B−1C

)C∗B−1

= B + B−1CC∗ + CC∗B−1 + B−1(CC∗B−1)2

+ B−1C(E − C∗B−1C

)C∗B−1

= B + B−1CC∗ + CC∗B−1 + B−1CEC∗B−1

= X11,

(L∗LDL∗L

)12 = L∗

11L11D11L∗21L22 + L∗

21L21D11L∗21L22 + L∗

21L22D22L∗22L22

= D11L∗21 + L∗

21L21D11L∗21 + L∗

21D22

= BB−1C + B−1CC∗B−1BB−1C + B−1C(E − C∗B−1C

)

= C + B−1CC∗B−1C + B−1CE − B−1CC∗B−1C

= C + B−1CE = X12,

(L∗LDL∗L

)21 = L∗

22L21D11L∗11L11 + L∗

22L21D11L∗21L21 + L∗

22L22D22L∗22L21

= L21D11 + L21D11L∗21L21 + D22L21

= C∗B−1B + C∗B−1BB−1CC∗B−1 + (E − C∗B−1C

)C∗B−1

= C∗ + C∗B−1CC∗B−1 + (E − C∗B−1C

)C∗B−1

= C∗ + EC∗B−1 = X∗12,

(L∗LDL∗L

)22 = L∗

22L21D11L∗21L22 + L∗

22L22D22L∗22L22

= L21D11L∗21 + D22

= C∗B−1BB−1C + E − C∗B−1C

= E = X22.

582 I. Stanimirovic

According to Lemma 1.1, the inverse matrix X−1 is equal to the block matrix Y =[ Y11 Y12

Y21 Y22

]. Then the following equations are satisfied:

LX−1L∗ =[

L11Y11L∗11 L11Y11L

∗21 + L11Y12L

∗22

L21Y11L∗11 + L22Y21L

∗21 (L21Y21 + L22Y21)L

∗21 + (L21Y12 + L22Y22)L

∗22

]

=[

Y11 Y11L∗21 + Y12

L21Y11 + Y21 (L21Y11 + Y21)L∗21 + L21Y12 + Y22

]

,

A∗AA∗ =[(A∗

11A11 + A∗21A21)A

∗11 + (A∗

11A12 + A∗21A22)A

∗12

(A∗12A11 + A∗

22A21)A∗11 + (A∗

12A12 + A∗22A22)A

∗12

(A∗11A11 + A∗

21A21)A∗21 + (A∗

11A12 + A∗21A22)A

∗22

(A∗12A11 + A∗

22A21)A∗21 + (A∗

12A12 + A∗22A22)A

∗22

]

=[Σ11 Σ12Σ21 Σ22,

]

.

From the expression (4.14) of Theorem 4.2 we have that A† = LX−1L∗ · A∗AA∗.As the equality L21 = M holds, the expression (4.18) holds from the product of theblock matrices LX−1L∗ and A∗AA∗. �

For the given block matrix A = [ A11 A12A21 A22

], the blocks of the partitioned matrix

(A∗A)2 = [B CC∗ E

]can be easily calculated by using the notation

αij = A∗1iA1j + A∗

2iA2j , i, j = 1,2.

Therefore, the blocks B,C,E can be evaluated as:

B = α211 + α12α21,

C = α11α12 + α12α22,

E = α21α12 + α222.

The calculation of the basic matrices M = C∗B−1 and N = CM leads to theevaluation of the blocks of the matrix X, using (4.16). The inverse matrix X−1 needsto be calculated. Notice that the blocks of Σ can be evaluated by:

Σ11 = α11A∗11 + α12A

∗12,

Σ12 = α11A∗21 + α12A

∗22,

Σ21 = α21A∗11 + α22A

∗12,

Σ22 = α21A∗21 + α22A

∗22.

Finally, the Moore-Penrose inverse A† can be determined according to (4.18).


Example 4.1 Consider the following 2 × 2 block matrix:

A =

⎡

⎢⎢⎢⎢⎢⎢⎣

5 6 15 −616 9 6 97 10 7 13

−5 6 1 −6−6 −9 −6 −197 3 −7 12

⎤

⎥⎥⎥⎥⎥⎥⎦

.

According to Theorem 4.3, the blocks B,C and E of the matrix (A∗A)2 are as fol-lows:

B =[

505414 427211427211 384895

]

, C =[

276196 664563272679 549944

]

,

E =[

267870 271621271621 997375

]

,

and the block matrices X11, X12 and X22 have the form:

X11 =[

684308607306455887247574383144530479875286494081

135121448833328319117306197144530479875286494081

135121448833328319117306197144530479875286494081

183286313055058544078250686144530479875286494081

]

,

X12 =[

625409183900745512022083009

28903445814078571335787001

696918789917389412022083009

6724343951473191335787001

]

, X22 =[

267870 271621271621 997375

]

.

The inverse of the partitioned matrix X can be easily calculated according to Corol-lary 3.1 and has the following form:

Y =

⎡

⎢⎢⎢⎣

906448559313191432836657050760392 − 10229238501285

1432836657050760392

− 102292385012851432836657050760392

191635092581707716418328525380196

66882803195127683929730317225681229402306559193379528 − 5280839149740617102018467

8612840614701153279596689764

− 248423624633565911913666517225681229402306559193379528

81749727214458014224055917225681229402306559193379528

66882803195127683929730317225681229402306559193379528 − 2484236246335659119136665

17225681229402306559193379528

− 52808391497406171020184678612840614701153279596689764

81749727214458014224055917225681229402306559193379528

54401900272158844806480688325880253834973511600883350197925569793652588 − 3706857351834826612842527539209653

23009841069605300101187553418761915528

− 370685735183482661284252753920965323009841069605300101187553418761915528

768061725192277262130902829032307123009841069605300101187553418761915528

⎤

⎥⎥⎥⎦

.

After performing several more calculations, the Moore-Penrose inverse of the matrixA is:

A† = 1

846414986

×⎡

⎣10488670 51081296 −14713647 −25185490 31824819 20669699−5934145 14986696 18308660 93274281 7854044 2503123432955519 −18409038 8264571 −38970491 −26726457 −40470883

−12297891 −22045404 12559500 −28255629 −36841766 −5147046

⎤

⎦.

584 I. Stanimirovic

Example 4.2 Let us now consider the rank-deficient matrix A2 from Example 2.2and its full-rank block LDL∗ decomposition. Moore-Penrose inverse of the constantmatrix L can be easily computed as

L† =[

1121

821

521

221 − 1

21 − 421

821

31105

22105

13105

4105 − 1

21

]

,

and since m = 1 we have that D−1 = D−111 = [ −8−x 9+x

9+x −10−x

]. According to Theo-

rem 4.1 the generalized inverse of A2 can be expressed by:

A†2 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1147 (−8 − 3x) 1

735 (−17 − 9x) 2−x245

1735 (29 + 3x)

1735 (−17 − 9x) −10−9x

122525−9x3675

80+9x3675

2−x245

25−9x3675

20−3x3675

5+x1225

1735 (29 + 3x) 80+9x

36755+x1225

−50−3x3675

1735 (52 + 9x)

9(5+x)1225

10+9x3675

−115−9x3675

5+x49

1735 (38 + 9x) 1

735 (1 + 3x) 1245 (−12 − x)

1735 (52 + 9x) 5+x

499(5+x)

12251

735 (38 + 9x)

10+9x3675

1735 (1 + 3x)

−115−9x3675

1245 (−12 − x)

−80−9x1225

1735 (−73 − 9x)

1735 (−73 − 9x) 1

147 (−22 − 3x)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Notice that this approach for Hermitian matrices is very beneficial and efficient, sincethe inverse of the block diagonal matrix D can be easily computed by determiningthe inverses of diagonal matrices Dii , i = 1,m.

Example 4.3 For the block Hermitian matrix F6 from Example 2.3, the expression(4.13) of Theorem 4.1 can be considered to determine the Moore-Penrose inverse F

†6 .

The block matrices L and D obtained earlier can be used to evaluate F†6 as

F†6 = (

L†)∗D−1L†

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 00 1 −4−a

5+a−3−a5+a

−2−a5+a

0 0 1 0 00 0 0 5

613

0 0 0 13

13

0 0 0 − 16

13

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

·

⎡

⎢⎢⎢⎢⎢⎣

1 −1 0 0 0−1 6+a

5+a0 0 0

0 0 2 −1 00 0 −1 −a 2 + a

0 0 0 2 + a −3 − a

⎤

⎥⎥⎥⎥⎥⎦


·

⎡

⎢⎢⎢⎢⎢⎣

1 0 0 0 0 00 1 0 0 0 0

0 −4−a5+a

1 0 0 0

0 −3−a5+a

0 56

13 − 1

6

0 −2−a5+a

0 13

13

13

⎤

⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −1 0 0 0 0−1 2 −1 0 0 0

0 −1 2 − 56 − 1

316

0 0 − 56

79 − a

449

19 + a

4

0 0 − 13

49

19 − 2

9

0 0 16

19 + a

4 − 29 − 5

9 − a4

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

5 Conclusion

A method for block LDL∗ decomposition of a constant block matrix is described andapplied to find the inverse and the Moore-Penrose inverse of a 2×2 block matrix. Themethod for the calculation of generalized inverses of a given rational matrix, based onthe LDL∗ factorization, was introduced in [9]. By proceeding with this recent result,we developed a method for the direct calculation of the sub-matrices occurring inMoore-Penrose inverse.

Extending and generalizing these results to the case of rational matrices and the2-variables case is the motivation for future research. Also, other types of generalizedinverses can be computed using the full-rank block LDL∗ decomposition.

Acknowledgements The author gratefully acknowledges support from the Research Project 174013 ofthe Serbian Ministry of Education and Science.

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586 I. Stanimirovic

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Documents

Full-rank block LDL∗ decomposition and the inverses of n×n block matrices