[PPT] - General inverses of matrices Ivana M. Staniev University of PowerPoint Presentation

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General inverses of matrices

Ivana M. Stanišev

University of Belgrade, Technical faculty in Bor Vienna, December 17 – 20, 2016.

1 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Indefinite inner products

We consider the space Cn with an indefinite inner product [.,.] induced by Hermitian matrix H ∈ Cn×n via the formula [x, y] = Hx, y, x, y ∈ Cn (1) where ., . denotes the standard inner product on Cn. If the Hermitian matrix H is invertible, then the indefinite inner product is nondegenerate, i.e. if [x, y] = 0 for every y ∈ Cn, then x = 0. For every matrix T ∈ Cn×n there is the unique matrix T[∗] satisfying [T[∗]x, y] = [x, Ty], for all x, y ∈ Cn, (2) and it is given by T[∗] = H−1T∗H. In more general case, for some matrix T ∈ Cm×n, the adjoint of a matrix T is given by T[∗] = N−1T∗M, (3) where M and N are Hermitian matrices that induce indefinite inner products on Cm and Cn, respectively.

2 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Spaces with a degenerate inner product (that is, those whose Gram matrix H is singular) often appear in applications. (e.g. in the theory of operator pencils). There exists a vector x ∈ Cn\{0} such that [x, y] = 0 for all y ∈ Cn. A problem - the H-adjoint of the matrix T ∈ Cn×n need not exist or if it exists it need not be unique; concept of linear relations; A matrix T ∈ Cn×n can always be interpreted as a linear relation via its graph Γ(T), where: Γ(T) :=

x

Tx

: x ∈ Cn
⊆ C2n. We will consider H-adjoint T[∗]

not as a matrix, but as a linear relation in Cn, i.e. a subspace of C2n. The H-adjoint of T is the linear relation T[∗] =

y

z

∈ C2n : [y, ω] = [z, x] for all
x

ω

∈ T
.

(4) We can always find a basis of Cn such that the matrices H and T have the forms: H =

H1
and T =
T1

T2 T3 T4

,

where H1, T1 ∈ Cm×m, m ≤ n, and H1 is invertible.

3 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Here H1 is an invertible Hermitian matrix and the inner product induced by it is

nondegenerate. We have

T[∗]H =                        y1 y2 T1[∗]H1 y1 z2             : T2∗H1y1 = 0            .

4 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Moore-Penrose inverse in nondegenerate IIPS

K. Kamaraj, K. C. Sivakumar, Moore-Penrose inverse in an indefinite inner product

space, J. Appl. Math. & Computing, 19 (1-2) (2005), 297-310; the existence is not guaranteed; they gave necessary and sufficient conditions for the existence of the Moore-Penrose inverse; if this inverse exists, then it is unique; if the indefinite inner products on Cn and Cm are induced by positive definite matrices N and M, then this inverse coincide with the weighted Moore-Penrose inverse; they gave some properties which are analogue to some of the Moore-Penrose inverse in Euclidean space.

5 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Definition: Moore-Penrose inverse Let A ∈ Cm×n. A matrix X ∈ Cn×m that satisfies the following four equations AXA = A, (1) XAX = X, (2) (AX)[∗] = AX, (3) (XA)[∗] = XA, (4) is called the Moore-Penrose inverse of a matrix A and it is denoted by X = A[†]. Theorem: Let A ∈ Cn×m. Then A[†] exists if and only if rank(AA[∗]) = rank(A[∗]A) = rank(A). (5) If A[†] exists then it is unique.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Theorem: Properties of the Moore-Penrose inverse Let λ ∈ C. If far a matrix A ∈ Cm×n the Moore-Penrose inverse exists A[†] ∈ Cn×m, then the next properties hold: (i) (A[†])[†] = A; (ii) (A[†])[∗] = (A[∗])[†]; (iii) (λA)[†] = λ[†]A[†], where λ[†] =

λ−1,

λ 0, 0, λ = 0; (iv) A[∗] = A[∗]AA[†] i A[∗] = A[†]AA[∗]; (v) (A[∗]A)[†] = A[†](A[†])[∗] i (AA[∗])[†] = (A[∗])[†]A[†]; (vi) A[†] = (A[∗]A)[†]A[∗] = A[∗](AA[∗])[†].

7 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

in degenerate inner product spaces (H is singular) the third and the fourth condition are never satisfied;

n the left side there is not a matrix but a linear relations;

notice that these conditions are equivalent to the fact that AX and XA are H-symmetric linear relations; we modify their definition so the new one consider not just matrices but also linear relations. Definition: Let A ⊆ C2n be a linear relation. A linear relation X ⊆ C2n is a Moore-Penrose inverse

f A if it satisfies the following four conditions:

AXA = A (1) XAX = X (2) AX ⊆ (AX)[∗] (3) XA ⊆ (XA)[∗] (4) A Moore-Penrose inverse of A is denoted by A[†].

8 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Moore-Penrose inverse in degenerate IIPS

I. M. Radojevi´

c, D. S. Djordjevi´ c: Moore-Penrose inverse in indefinite inner product spaces, Filomat. we define a Moore-Penrose inverse of matrices and linear relations in indefinite inner product spaces; for an invertible Hermitian matrix H that induces the indefinite inner product, this inverse coincide to the one that Kamaraj and Sivakumar defined; if a matrix H is positive definite, this inverse coincide to the weighted Moore-Penrose inverse; if H = I, this inverse coincide to the one in Euclidean spaces.

9 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

A moore-Penrose inverse need not exist; if it exists, it does NOT have to be unique; if for some square matrix a Moore-Penrose inverse exists,we give some properties that are similar to the ones in nondegenerate case (given by Kamaraj and Sivakumar); we consider Moore-Penrose inverse of H-normal matrix; we give a representation for a Moore-Penrose inverse for the special case - when the H-adjoint of a matrix A has a full domain; Representation of a Moore-Penrose inverse of H-normal matrix with some additional conditions. H =

H1
, A =
A1

A2 A3 A4

and X =
X1

X2 X3 X4

where H, A, X ∈ Cn×n.

10 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Theorem: Let A ∈ Cn×n and H ∈ Cn×n be Hermitian matrices. A matrix X ∈ Cn×n is a Moore-Penrose inverse of a matrix A if the following four equations are satisfied: (1) AXA = A, (2) XAX = X, (3) (HAX)∗ = HAX, (4) (HXA)∗ = HXA. Example: Let A =

1

1

and H =
1
. A matrix X =
1

c

is a Moore-Penrose

inverse of A, where a c is an arbitrary complex number. If a Moore-Penrose inverse exists, it is not unique.

11 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Theorem: Let A ∈ Cn×n. Then a matrix X ∈ Cn×n is a Moore-Penrose inverse of A if and only if the following conditions hold: (i) A1X2 + A2X4 = 0, (ii) A1X1 + A2X3 je H1-selfadjoint matrix, (iii) X1A2 + X2A4 = 0, (iv) X1A1 + X2A3 je H1- selfadjoint matrix, (v) A1X1A1 + A2X3A1 = A1, (vi) A1X1A2 + A2X3A2 = A2, (vii) A3X1A1 + A4X3A1 + A3X2A3 + A4X4A3 = A3, (viii) A4X3A2 + A4X4A4 = A4, (ix) X1A1X1 + X2A3X1 = X1, (x) X1A1X2 + X2A3X2 = X2, (xi) X3A1X1 + X4A3X1 + X3A2X3 + X4A4X3 = X3, (xii) X4A3X2 + X4A4X4 = X4.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

The properties of a Moore-Penrose inverse of a matrix in degenerate IIPS

Theorem: If A[†] ∈ Cn×n is a MoorePenrose inverse of a matrix A ∈ Cn×n, then (A[†])[∗] ⊆ (A[∗])[†]. In nondegenerate case (A[†])[∗] = (A[∗])[†] holds. Theorem: If A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n, then A[∗] = A[∗]AA[†] and A[†]AA[∗] ⊆ A[∗]. (6) In nondegenerate case A[∗] = A[∗]AA[†] and A[†]AA[∗] = A[∗] holds.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

The property (6) follows directly from the forms of the suitable linear relations: A[†]AA[∗] =                           y1 y2 A[∗]

1 y1

(X3A1 + X4A3)A[∗]

1 y1 + (X3A2 + X4A4)z2

             : A∗

2H1y1 = 0

             ⊆                          y1 y2 A[∗]

1 y1

z2             : A∗

2H1y1 = 0

             = A[∗]. The opposite inclusion obviously holds just in the case of the invertible product X3A2 + X4A4. Corollary: If in the previous Theorem A4 is an invertible matrix, then A[∗] = A[∗]AA[†] and A[†]AA[∗] = A[∗] holds.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Theorem: If A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n, then A[†](A[†])[∗] is an {1,2,(3)}- inverse of a linear relation A[∗]A. Theorem: If A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n, then (A[†])[∗]A[†] is an {1,2,(4)} - inverse of a linear relation AA[∗]. In nondegenerate case: (A[∗]A)[†] = A[†](A[†])[∗] and (AA[∗])[†] = (A[†])[∗]A[†] holds.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Theorem: If X = A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n then A[†](A[†])[∗] is a Moore-Penrose inverse of a linear relation A[∗]A if and only if X2∗H1y1 = 0 for all y =

y1

y2

that satisfy A2∗H1(A1y1 + A2y2) = 0.

Theorem: If X = A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n then (A[†])[∗]A[†] is a Moore-Penrose inverse of a linear relation AA[∗] if and only if A2∗H1y1 = 0 for all y =

y1

y2

that satisfy X2∗H1(X1y1 + X2y2) = 0.

Corollary: If a matrix X = A[†] ∈ Cn×n is a Moore-Penrose inverse of a matrix A ∈ Cn×n and A2 = 0 i X2 = 0 (i.e. linear relations A[∗] and X[∗] have full domains), then (A[∗]A)[†] = A[†](A[†])[∗] and (AA[∗])[†] = (A[†])[∗]A[†] holds.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

Open problem

To give necessary and sufficient conditions for the existence of a Moore-Penrose inverse for (square) matrices in degenerate indefinite inner product spaces.

17 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices Indefinite inner products Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate

(A[†])[∗] ⊆ (A[∗])[†] A[∗]AA[†] = A[∗] and A[†]AA[∗] ⊆ A[∗] A[†](A[†])[∗] is the {1, 2, (3)}-inverse of A[∗]A (A[†])[∗]A[†] is the {1, 2, (4)}-inverse of AA[∗] necessary and sufficient conditions are given such that A[†](A[†])[∗] is the Moore-Penrose inverse of A[∗]A and that (A[†])[∗]A[†] is the Moore-Penrose inverse of AA[∗] If exists A[†] is not unique Problem: the existence of a Moore-Penrose inverse

18 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices H-normal matrices

H-normal matrices

Let A be H-normal matrix. That means AA[∗] ⊆ A[∗]A. We have that AA[∗] =                          y1 y2 A1A1[∗]y1 + A2z2 A3A[∗]

1 y1 + A4z2

            : A∗

2H1y1 = 0

             , and A[∗]A =                       y1 y2 A1[∗](A1y1 + A2y2) z2            : A∗

2H1(A1y1 + A2y2) = 0

           . As it is well known, AA[∗] ⊆ A[∗]A if and only if A2 = 0 and A1A[∗]

1

= A[∗]

1 A1, i.e. that A1

is H1-normal. The opposite inclusion stands if A4 is invertible in addition. Here, we assume A ∈ Cn×n and A1 ∈ Cm×m, where m ≤ n. In general case, if the Moore-Penrose inverse for matrix A exists, it may happen that its submatrix A1 does not have the Moore-Penrose inverse.

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices H-normal matrices

Example Let H =         1 −1         . The Moore-Penrose inverse for A =         1 1 1 1 1 1         is the matrix X =         1 1 −1 −1 1         . But rank(A1) rank(A1A1[∗]), so the Moore-Penrose inverse for A1 does not exist. Theorem: Let A =

A1

A2 A3 A4

be an H-normal matrix. If A[†] ∈ Cn×n exists then A1[†] exists and

A1 is H1-range hermitian. Corollary If matrix A is H-normal and A[†] ∈ Cn×n is a matrix, then A1 is H1-EP matrix.

20 Ivana M. Stanišev General inverses of matrices

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices H-normal matrices

Moore-Penrose inverse of H-normal matrix

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Moore-Penrose inverse Moore-Penrose inverse for H − normal matrices H-normal matrices

Thank you!

23 Ivana M. Stanišev General inverses of matrices