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Introduction A new formulation of the nullity theorem A formula for the Moore-Penrose inverse Summary Generalized inverses of banded matrices Enrico Bozzo joint work with R. Bevilacqua, G. Del Corso, D. Fasino Dipartimento di Informatica, Università di Pisa INTERNATIONAL SEMINAR ON MATRIX METHODS AND OPERATOR EQUATIONS June 20 - 25, 2005, Moscow, Russia Bozzo et al. Generalized inverses of banded matrices
Transcript

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inverses of banded matrices

Enrico Bozzojoint work with R. Bevilacqua, G. Del Corso, D. Fasino

Dipartimento di Informatica, Università di Pisa

INTERNATIONAL SEMINAR ON MATRIX METHODS ANDOPERATOR EQUATIONS

June 20 - 25, 2005, Moscow, Russia

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Banded matrices

Banded matrices are basic in matrix theory and inapplications and many authors studied the structure oftheir inverses.By assuming that the entries in the outermost diagonalsare nonzero it is possible to derive inversion formulasinvolving Schur complements.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Banded matrices

Banded matrices are basic in matrix theory and inapplications and many authors studied the structure oftheir inverses.By assuming that the entries in the outermost diagonalsare nonzero it is possible to derive inversion formulasinvolving Schur complements.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Schur complement

Let B be a n × n nonsingular banded matrix of the form

B =

(B11 LB21 B22

)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then

B =

(Iq O

B22L−1 Ip

) (O LS O

) (Ip O

L−1B11 Iq

),

where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Schur complement

Let B be a n × n nonsingular banded matrix of the form

B =

(B11 LB21 B22

)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then

B =

(Iq O

B22L−1 Ip

) (O LS O

) (Ip O

L−1B11 Iq

),

where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Schur complement

Let B be a n × n nonsingular banded matrix of the form

B =

(B11 LB21 B22

)where L is q × q lower triangular and nonsingular and p = n− qis the bandwidth. Let Ik be the k × k identity. Then

B =

(Iq O

B22L−1 Ip

) (O LS O

) (Ip O

L−1B11 Iq

),

where the p × p matrix S = B21 − B22L−1B11 is known as Schurcomplement of L in B and is nonsingular.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Example of inversion formula

This implies

B−1 =

(0 0

L−1 0

)+

(Ip

−L−1B11

)S−1 (

−B22L−1 Ip).

The submatrices of B−1 contained in the part of the matrix not“covered” by L−1 have rank not greater than the bandwidth p.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Example of inversion formula (continued)

If p = 1 then B is an unreduced lower Hessenberg and

B−1 =

(0 0

L−1 0

)+

1S

(1

−L−1B11

) (−B22L−1 1

).

The upper triangular part of B−1 is the upper triangular part of amatrix whose rank is one.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Asplund, 1959

Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Asplund, 1959

Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Asplund, 1959

Edgar Asplud in 1959 published the first result withoutassumptions besides nonsingularity.In particular, Asplund showed that if a nonsingular matrixcontains a null submatrix then its complementarysubmatrix in the inverse has a prescribed rank.The notion of complementary submatrices leads us todiscuss the nullity theorem that can be seen as ageneralization of Asplund’s result.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices

Let A be a square matrix of order n, let N = {1, 2, . . . , n} and letα and β be nontrivial subsets of N. Then we denote withA(α, β) the submatrix of A having row indices in α and columnindices in β.

DefinitionLet A and B be square matrices of order n. Then the twosubmatrices A(α, β) and B(N \ β, N \ α) are said to becomplementary.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 1

In the matrix

A =

X X Y Y Y YX X Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrix A({1, 2}, {1, 2})is in red

In the matrix

B =

Y Y Y Y Y YY Y Y Y Y YY Y X X X XY Y X X X XY Y X X X XY Y X X X X

the submatrixB({3, 4, 5, 6}, {3, 4, 5, 6}) is inred

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 1

In the matrix

A =

X X Y Y Y YX X Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrix A({1, 2}, {1, 2})is in red

In the matrix

B =

Y Y Y Y Y YY Y Y Y Y YY Y X X X XY Y X X X XY Y X X X XY Y X X X X

the submatrixB({3, 4, 5, 6}, {3, 4, 5, 6}) is inred

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 2

In the matrix

A =

Y Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixA({1}, {3, 4, 5, 6}) is in red

In the matrix

B =

Y X X X X XY X X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixB({1, 2}, {2, 3, 4, 5, 6}) is in red

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 2

In the matrix

A =

Y Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixA({1}, {3, 4, 5, 6}) is in red

In the matrix

B =

Y X X X X XY X X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixB({1, 2}, {2, 3, 4, 5, 6}) is in red

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 3

In the matrix

A =

Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixA({1, 2}, {3, 4, 5, 6}) is in red

In the matrix

B =

Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixB({1, 2}, {3, 4, 5, 6}) is in red

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Complementary submatrices: example 3

In the matrix

A =

Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixA({1, 2}, {3, 4, 5, 6}) is in red

In the matrix

B =

Y Y X X X XY Y X X X XY Y Y Y Y YY Y Y Y Y YY Y Y Y Y YY Y Y Y Y Y

the submatrixB({1, 2}, {3, 4, 5, 6}) is in red

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

The nullity theorem

Let null(·) denote the nullity, i.e., the dimension of the nullspace.

TheoremLet A be n × n and nonsingular. Then

null(A(α, β)) = null(A−1(N \ β, N \ α))

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Dr. Nullity and Mr. Rank

Let rk(·) denote the rank. For a matrix, not necessarily square,the sum of the rank and of the nullity equals the number ofcolumns.

CorollaryLet A be n × n and nonsingular. Then

rk(A−1(N \ β, N \ α)) = rk(A(α, β)) + n − (|α|+ |β|).

In the case where A(α, β) = O the corollary is exactly Asplundresult.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Application to banded matrices

If B is a nonsingular banded matrix with bandwidth p if wechoose

α = {1, . . . , k}β = {p + 1 + k , . . . , n} k = 1, . . . , n − p − 1

then B(α, β) = O and n − (|α|+ |β|) = p so that

rk(B−1(N \ β, N \ α)) = p.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

Definition

DefinitionLet A be a square matrix of order n, let N = {1, 2, . . . , n} and letΣ ⊆ N × N. The structured rank rk(A,Σ) is defined as themaximum of rk(A(α, β)) s.t. α× β ⊆ Σ.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Inverses of banded matricesThe nullity theoremStructured rank

An example of invariance under inversion

Let Σu = {(i , j)|i , j ∈ N ∧ j > i}. The “rank” version of the nullitytheorem allows to prove, for example, that

rk(A,Σu) = rk(A−1,Σu)

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

The Penrose Equations

1 AXA = A2 XAX = X3 (AX )∗ = AX4 (XA)∗ = XA

DefinitionGiven a matrix A let A{j , . . . , k} be the set of matrices whichsatisfy equations j , . . . , k among the four above. A matrixX ∈ A{j , . . . , k} is called an {j , . . . , k}-inverse of A.

For example A{1, 2, 3, 4} has exactly one element known asMoore-Penrose inverse of A.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

{1, 2}-inverses

If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

{1, 2}-inverses

If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

{1, 2}-inverses

If X ∈ A{1, 2} then:A ∈ X{1, 2};rk(X ) = rk(A);the nullspace of A and the range of X are supplementarysubspaces.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

A nullity theorem for {1, 2} inverses

TheoremLet A be n × n and X ∈ A{1, 2}. Then

|null(X (N \ β, N \ α))− null(A(α, β))| ≤ null(A) = null(X ).

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

The rank counterpart

Corollary

Let A be n × n and X ∈ A{1, 2}. Then

rk(X (N \ β, N \ α)) ≤ rk(A(α, β)) + n − (|α|+ |β|) + null(A)

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Application to banded matrices

If A is a banded matrix with bandwidth p and we choose

α = {1, . . . , k}β = {p + 1 + k , . . . , n} k = 1, . . . , n − p − 1

then A(α, β) = O and n − (|α|+ |β|) = p so that if X ∈ A{1, 2}then

rk(X (N \ β, N \ α)) ≤ p + null(A).

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Example

The matrix

Z =

0 0 0 01 0 0 00 1 0 00 0 1 0

is banded with bandwidth p = 2, 1, 0,−1.The matrices in Z{1, 2} have the form

a 1 0 0b 0 1 0c 0 0 1

da + eb + fc d e f

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Example (generalization)

Let 1 ≤ p ≤ n − 1 and let Zp be n × n defined as

Zp =

(O O

In−p O

)then the matrices in Zp{1, 2} have the form(

X In−pYX Y

)

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Generalized inversesThe new formulationApplications

Application to structured rank

The extended formulation of nullity theorem shows, forexample, that if A is square and B ∈ A{1, 2} then

|rk(A,Σu)− rk(B,Σu)| ≤ null(A)

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Outline

1 IntroductionInverses of banded matricesThe nullity theoremStructured rank

2 A new formulation of the nullity theoremGeneralized inversesThe new formulationApplications

3 A formula for the Moore-Penrose inverse

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Assumptions

In order to simplify the presentation we assume that B is ann × n unreduced singular lower Hessenberg matrix. Recall that

B =

(In−1 O

B22L−1 1

) (O LS O

) (1 O

L−1B11 In−1

).

If B is singular S = 0 and setting P = L−1B11 and Q = B22L−1

we find

B =

(In−1Q

)L

(P In−1

).

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Formula for the Moore-Penrose inverse

B+ =

(O O

L−1 O

)+

P∗L−1Q∗

(1 + P∗P)(1 + QQ∗)

(1−P

) (−Q 1

)+

1(1 + P∗P)

(1−P

) (P∗L−1 O

)+

1(1 + QQ∗)

(O

L−1Q∗

) (−Q 1

).

The formula shows that the rank of the submatrices in theupper triangular part cannot exceed two.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Example

Let us consider the following tridiagonal matrix

T =

0 1 0 0 01 0 1 0 00 1 0 1 00 0 1 0 10 0 0 1 0

.

The matrix T is singular and null(T ) = 1.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Example (continued)

The Moore-Penrose inverse of T is

T + =13

0 2 0 −1 02 0 1 0 10 1 0 1 0

−1 0 −1 0 20 −1 0 2 0

.

The rank of every submatrix of T + above (or below) the maindiagonal is less or equal to 2.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Summary

We presented a new formulation of nullity theorem thatapplies to singular matrices.Generalized inverses of square banded matrices havestructured rank properties that recall those enjoyed byordinary inverses.We presented a formula for the Moore-Penrose inverse ofa singular banded matrix having nonzero outermostdiagonal.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Work in progress

Rectangular matricesA paper on this subject is under revision.What happens for infinite matrices?This would be an interesting research topic.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

Work in progress

Rectangular matricesA paper on this subject is under revision.What happens for infinite matrices?This would be an interesting research topic.

Bozzo et al. Generalized inverses of banded matrices

IntroductionA new formulation of the nullity theorem

A formula for the Moore-Penrose inverseSummary

References

E. AsplundInverses of matrices {aij} which satisfy aij = 0 for j > i + pMathematica Scandinavica, 7:57-60, 1959.

R. Bevilacqua, E. Bozzo, G. M. Del Corso, D. FasinoRank structure of generalized inverses of banded matricesReport, Dipartimento di Informatica, Università di Pisa,2005.

G. Strang, T. NguyenThe interplay of ranks of submatricesSIAM Review, 46:637-646, 2005.

Bozzo et al. Generalized inverses of banded matrices


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