H-matrix theory and applications Maja Nedovi University of Novi - - PowerPoint PPT Presentation

H-matrix theory and applications

Maja Nedović

University of Novi Sad, Serbia

joint work with Ljiljana Cvetković

MatTriad 2015, Coimbra

Contents

• H-matrices and SDD-property
• Benefits from H-subclasses
• Breaking the SDD
• Additive and multiplicative conditions
• Partitioning the index set
• Recursive row sums
• Nonstrict conditions

H-matrices and SDD-property

A complex matrix A=[aij]nxn is an SDD- matrix if for each i from N it holds that

aii > r

i A

( ) =

aij

j∈N, j≠i

∑

Lev Lévy-Desplanques: nonsingular Deleted row sums

H-matrices and SDD-property

A complex matrix A=[aij]nxn is an SDD- matrix if for each i from N it holds that Lev A complex matrix A=[aij]nxn is an H-matrix if and only if there exists a diagonal nonsingular matrix W such that AW is an SDD matrix.

aii > r

i A

( ) =

aij

j∈N, j≠i

∑

H-matrices and SDD-property

A complex matrix A=[aij]nxn is an SDD- matrix if for each i from N it holds that HH

SDD

H

aii > r

i A

( ) =

aij

j∈N, j≠i

∑

H-matrices and SDD-property

A complex matrix A=[aij]nxn is an SDD- matrix if for each i from N it holds that HH

SDD

H

?

aii > r

i A

( ) =

aij

j∈N, j≠i

∑

Subclasses of H-matrices & diagonal scaling characterizations Benefits:

• 1. Nonsingularity result covering a wider matrix class
• 2. A tighter eigenvalue inclusion area (not just for the observed class)
• 3. A new bound for the max-norm of the inverse for a wider matrix class
• 4. A tighter bound for the max-norm of the inverse for some SDD matrices
• 5. Schur complement related results (closure and eigenvalues)
• 6. Convergence area for relaxation iterative methods
• 7. Sub-direct sums
• 8. Bounds for determinants

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions Ostrowski, Pupkovi Ostrowski, A. M. (1937), Pupkov, V. A. (1983), Hoffman, A.J. (2000), Varga, R.S. : Geršgorin and his circles (2004)

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions Ostrowski, Pupkovi S-SDD, PH Gao, Y.M., Xiao, H.W. (1992), Varga, R.S. (2004), Dashnic, L.S., Zusmanovich, M.S. (1970), Kolotilina, l. Yu.(2010), Cvetković, Lj., Nedović, M. (2009), (2012), (2013).

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions Ostrowski, Pupkovi S-SDD, PH Mehmke, R., Nekrasov, P. (1892), Gudkov, V.V. (1965), Szulc, T. (1995), Li, W. (1998), Cvetković, Lj., Kostić, V., Nedović, M. (2014). Nekrasov- matrices

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions Ostrowski, Pupkovi S-SDD, PH

• O. Taussky (1948), Beauwens (1976), Szulc,T. (1995), Li, W. (1998),

Varga, R.S. (2004) Cvetković, Lj., Kostić, V. (2005) Nekrasov- matrices IDD, CDD S-IDD, S-CDD

Breaking the SDD

SDD

Additive and multiplicative conditions Partitioning the index set Recursive row sums Non-strict conditions Ostrowski, Pupkovi S-SDD, PH

• O. Taussky (1948), Beauwens (1976), Szulc,T. (1995), Li, W. (1998),

Varga, R.S. (2004) Cvetković, Lj., Kostić, V. (2005) Nekrasov- matrices IDD, CDD S-IDD, S-CDD

H

( ) ( )

A r A r a a

j i jj ii

>

I Additive and multiplicative conditions

( ) ( ) ( )

A r A r a a A r a a

j i jj ii i ji i j ii

+ > + >

≠

} }, { min{max Ostrowski-matrices multiplicative condition: Pupkov-matrices additive condition:

Ostrowski, A. M. (1937), Pupkov, V. A. (1983), Hoffman, A.J. (2000), Varga, R.S. : Geršgorin and his circles (2004)

II Partitioning the index set

( )

S i a A r a

i j S j ij S i ii

∈ = >

∑

≠ ∈

,

,

( )

( )

( )

( )

( ) ( )

S j S i A r A r A r a A r a

S j S i S j jj S i ii

∈ ∈ > − − , ,

S S

S-SDD-matrices

Given any complex matrix A=[aij]nxn and given any nonempty proper subset S of N, A is an S-SDD matrix if Gao, Y.M., Xiao, H.W. LAA (1992) Cvetković, Lj., Kostić, V., Varga, R. ETNA (2004)

• A matrix A=[aij]nxn is an S-SDD matrix iff there exists a matrix W in Ws

such that AW is an SDD matrix.

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∈ = ∈ > = = = S i for w and S i for w w w w diag W

i i n S

1 : ,..., ,

2 1

γ W S N\S 1

SDD

γ γ 1 1 . . . .

S-SDD-matrices

II Partitioning the index set

Diagonal scaling characterization & Scaling matrices

S N\S 1

SDD

γ γ 1 1 . . . .

We choose parameter from the interval:

( ) ( ) ( ),

,

2 1

A A I γ γ

γ =

( ) ( ) ( ),

max

1

A r a A r A

S i ii S i S i

− = ≤

∈

γ

( ) ( ) ( )

. min

2

A r A r a A

S j S j jj S j

− =

∈

γ

Diagonal scaling characterization & Scaling matrices

SDD

S-SDD

Diagonal scaling characterization & Scaling matrices

SDD

S-SDD

γ γ γ γ γ γ γ

T-SDD

Diagonal scaling characterization & Scaling matrices

SDD

S-SDD

γ γ γ γ γ γ γ

T-SDD ∑-SDD

Diagonal scaling characterization & Scaling matrices

SDD

γ γ γ γ γ γ γ

∑-SDD

Eigenvalue localization

( ) ( )

{ }

, , : S i A r a z C z A

S i ii S i

∈ ≤ − ∈ = Γ

( ) ( )

( )

( )

( )

( ) ( )

{ }

. , , : S j S i A r A r A r a z A r a z C z A V

S j S i S j jj S i ii S ij

∈ ∈ ≤ − − − − ∈ =

( ) ( ) ( ) ( ) .

,

  

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ = ⊆

∈ ∈ ∈ S j S i S ij S i S i S

A V A A C A σ

Cvetković, L., Kostić, V., Varga R.S.: A new Geršgorin-type eigenvalue inclusion set ETNA, 2004. Varga R.S.: Geršgorin and his circles, Springer, Berlin, 2004.

Schur complement

Schur The Schur complement of a complex nxn matrix A, with respect to a proper subset α of index set N={1, 2,…, n}, is denoted by A/ α and defined to be: ( )

( ) ( ) ( ) ( )

α α α α α α , ,

1 A

A A A

−

−

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

SDD Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979), 246-251.

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

SDD Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979), 246-251.

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

Ostr

Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979)

Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997)

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

Ostr

Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979)

Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997)

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

$

Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979)

Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997) Zhang, F. : The Schur complement and its applications, Springer, NY, (2005).

Schur complement

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

$

Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices.

• Czech. Math. J. 29 (104) (1979)

Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997) Zhang, F. : The Schur complement and its applications, Springer, NY, (2005).

Schur complements of S-SDD

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

∑-SDD

Cvetković, Lj., Kostić, V., Kovačević, M., Szulc, T. : Further results on H-matrices and their Schur complements. AMC (2008) Liu, J., Huang, Y., Zhang, F. : The Schur complements of generalized doubly diagonally dominant matrices. LAA (2004)

Schur complements of S-SDD

Schur

( )

α A

( )

α α, A

( )

α α , A

( )

α A

( )

α A

( )

α α, A

∑-SDD

Cvetković, Lj., Kostić, V., Kovačević, M., Szulc, T. : Further results on H-matrices and their Schur complements. AMC (2008) Liu, J., Huang, Y., Zhang, F. : The Schur complements of generalized doubly diagonally dominant matrices. LAA (2004)

Schur complements of S-SDD

Cvetković, Lj., Kostić, V., Kovačević, M., Szulc, T. : Further results on H-matrices and their Schur complements. AMC (2008)

• Theorem2. Let A=[aij]nxn be an S-SDD matrix. Then for any nonempty proper

subset α of N such that S is a subset of α or N\S is a subset of α, A/ α is an SDD matrix.

• Theorem1. Let A=[aij]nxn be an ∑-SDD matrix. Then for any nonempty proper

subset α of N, A/ α is also an ∑-SDD matrix. More precisely, if A is an S-SDD matrix, then A/ α is an (S\ α)-SDD matrix. Cvetković, Lj., Nedović, M. : Special H-matrices and their Schur and diagonal- Schur complements. AMC (2009)

Eigenvalues of the SC

σ A

( ) ⊆ Γ A ( ) =

Γi A

( )

i∈N



=

i∈N



z ∈ C : z − aii ≤ r

i A

( ) =

aij

j∈N \ i

{ }

∑

' ( ) * ) + , )

• )

( )

α A

( )

α α, A

( )

α α, A

( )

α A

( )

α σ λ / A ∈

Liu, J., Huang, Z., Zhang, J. : The dominant degree and disc theorem for the Schur complement. AMC (2010) SDD

( ) ( ),

max A r a A r

i ii i i α α α

γ − >

∈

( )

( ) ( ) ( )

( ) ( ).

, / /

1 1

A r A r R AW W AW W A

j j j j j α α α

γ α σ α σ + = Γ ⊆ =

∈ − −



. , α = − ∈ S SDD S A

Eigenvalues of the SC of S-SDD

Cvetković, Lj., Nedović, M. : Eigenvalue localization refinements for the Schur

• complement. AMC (2012)
• ­‑ 50

• ­‑ 50

• ­‑ 50

• ­‑ 50

Weighted Geršgorin set for the Schur complement matrix Cvetković, Lj., Nedović, M. : Diagonal scaling in eigenvalue localization for the Schur complement. PAMM (2013)

Eigenvalues of the SC

• Let A be an SDD matrix with real diagonal entries and let α be a proper

subset of N. Then, A/α and A(N\α) have the same number of eigenvalues whose real parts are greater (less) than w (resp. -w), where

( ) ( ) ( ) ( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + − =

∈ ∈

A r a A r a A r a A w

j ii i ii i j jj j α α α

min min Liu, J., Huang, Z., Zhang, J. : The dominant degree and disc theorem for the Schur complement. AMC (2010)

SC

( )

α A

( )

α α, A

( )

α α, A

( )

α A

( )

α A

( )

α α, A

• Let A be an S-SDD matrix with real diagonal entries and let α be a proper

subset of N. Then, A/ α and A(N\α) have the same number of eigenvalues whose real parts are greater (less) than w (resp. -w), where

( )

AW W w w

1 −

=

Eigenvalues of the SC of S-SDD

Cvetković, Lj., Nedović, M. : Eigenvalue localization refinements for the Schur

• complement. AMC (2012)

Remarks:

• This result covers a wider class of matrices.
• By changing the parameter in the scaling matrix we obtain more vertical

bounds with the same separating property.

• We can apply it to an SDD matrix, observing that it belongs to T-SDD

class for any T subset of N.

SC

( )

α A

( )

α α, A

( )

α α, A

( )

α A

( )

α A

( )

α α, A

Eigenvalues of the SC of S-SDD

Dashnic-Zusmanovich

Dashnic, L. S., Zusmanovich, M.S.: O nekotoryh kriteriyah regulyarnosti matric i lokalizacii spectra. Zh. vychisl. matem. i matem. fiz. (1970) Dashnic, L. S., Zusmanovich, M.S.: K voprosu o lokalizacii harakteristicheskih chisel matricy. Zh. vychisl. matem. i matem. fiz. (1970) A matrix A=[aij]nxn is a Dashnic-Zusmanovich (DZ) matrix if there exists an index i in N such that

( )

( )

( )

. , , N j i j a A r a A r a a

ji i ji j jj ii

∈ ≠ > + −

1

SDD

γ 1 1 . .

PH-matrices

Kolotilina, L. Yu. : Diagonal dominance characterization of PM- and PH-

• matrices. Journal of Mathematical Sciences (2010)

n x n l x l Aggregated matrices

l

S S S ,..., , :

2 1

π

PH-matrices

Kolotilina, L. Yu. : Diagonal dominance characterization of PM- and PH-

• matrices. Journal of Mathematical Sciences (2010)

* * * *

Nekrasov matrices

• A complex matrix A=[aij]nxn is SDD-

matrix if for each i from N it holds that

• A complex matrix A=[aij]nxn is a

Nekrasov-matrix if for each i from N it holds that

( ) ( )

A r A d >

( ) ( ) ( ) ( ) ( )

. ,..., 3 , 2 , , ,

1 1 1 1 1

n i a a A h a A h A r A h A h a

n i j ij jj j i j ij i i ii

= + = = >

∑ ∑

+ = − =

( )

∑

≠ ∈

= >

i j N j ij i ii

a A r a

,

( ) ( )

A h A d >

U L D A − − =

Nekrasov matrices

A complex matrix A=[aij]nxn is a Nekrasov-matrix if for each i from N it holds that

( ) ( ) ( ) ( ) ( )

. ,..., 3 , 2 , , ,

1 1 1 1 1

n i a a A h a A h A r A h A h a

n i j ij jj j i j ij i i ii

= + = = >

∑ ∑

+ = − =

( ) ( )

A h A d >

U L D A − − =

Nekrasov row sums

Nekrasov matrices and scaling

• Theorem. Let A=[aij]nxn be a Nekrasov matrix with nonzero

Nekrasov row sums. Then, for a diagonal positive matrix D where

( )

, , , 1 , n i a A h d

ii i i i

… = = ε

and is an increasing sequence of numbers with

( )

n i i 1 =

ε

( )

, , , 2 , , 1 , 1

1

n i A h a

i ii i

… = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∈ = ε ε

the matrix AD is an SDD matrix. Szulc, T., Cvetković, Lj., Nedović, M. : Scaling technique for Nekrasov

• matrices. AMC (2015) (in print)

Nekrasov matrices and permutations

U L D A − − =

• Unlike SDD and H, Nekrasov class is NOT closed under

similarity (simultaneous) permutations of rows and columns!

• Given a permutation matrix P, a complex matrix A=[aij]nxn is called

P-Nekrasov if

• The union of all P-Nekrasov=

Gudkov class

( ) ( ) ( ) ( )

. , , AP P h AP P d N i AP P h AP P

T T T i ii T

> ∈ >

{P1,P2} – Nekrasov matrices

• Suppose that for the given matrix A=[aij]nxn and two given permutation

matrices P1 and P2 We call such a matrix {P1,P2} – Nekrasov matrix. A A1 A2

( ) ( ) ( )

{ }

( )

( )

. 2 , 1 , , , min

2 1

= = > k AP P h P A h A h A h A d

k T k k P P P

k

{P1,P2} – Nekrasov matrices

• 
• Theorem1. Every {P1, P2} – Nekrasov matrix is nonsingular.
• 
• Theorem2. Every {P1, P2} – Nekrasov matrix is an H – matrix.
• 
• Theorem3. Given an arbitrary set of permutation matrices

every Пn – Nekrasov matrix is nonsingular, moreover, it is an H – matrix.

{ }

p k k n

P

1 =

= Π

Cvetković, Lj., Kostić, V., Nedović, M. : Generalizations of Nekrasov matrices and applications. (2014)

Max-norm bounds for the inverse of {P1,P2} – Nekrasov matrices

• 
• Theorem1. Suppose that for a given set of permutation matrices {P1,

P2}, a complex matrix A=[aij]nxn, n>1, is a {P1, P2} – Nekrasov matrix. Then, where

( ) ( ) ( ) ( )

, , min 1 min , min max

2 1 2 1

1

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≤

∈ ∈ ∞ − ii P i ii P i N i ii P i ii P i N i

a A h a A h a A z a A z A

( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( )

( )

. , , , , ,..., 3 , 2 , 1 ,

1 1 1 1 1

AP P Pz A z A z A z A z n i a A z a A z A r A z

T P T n jj j i j ij i

= = = + = =

∑

− =

…

Cvetković, Lj., Kostić, V., Nedović, M. : Generalizations of Nekrasov matrices and applications. (2014)

• 
• Theorem2. Suppose that for a given set of permutation matrices {P1,

P2}, a complex matrix A=[aij]nxn, n>1, is a {P1, P2} – Nekrasov matrix. Then,

A−1

∞ ≤

max

i∈N

min zi

P

1 A

( ),zi

P

2 A

( )

{ }

% & ' ( min

i∈N

aii − min hi

P

1 A

( ),hi

P

2 A

( )

{ }

% & ' ( , ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( )

( )

. , , , , ,..., 3 , 2 , 1 ,

1 1 1 1 1

AP P Pz A z A z A z A z n i a A z a A z A r A z

T P T n jj j i j ij i

= = = + = =

∑

− =

…

Max-norm bounds for the inverse of {P1,P2} – Nekrasov matrices

Cvetković, Lj., Kostić, V., Nedović, M. : Generalizations of Nekrasov matrices and applications. (2014)

Numerical examples

• 

Observe the given matrix B and permutation matrices P1 and P2.

• 

Notice that B is a Nekrasov matrix.

Cvetković, Lj., Kostić, V., Doroslovački, K. : Max-norm bounds for the inverse of S- Nekrasov matrices. (2012)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

. 4464 . 40 , 857 . 117 , 25 . 101 , 45 , 25 . 41 , 607 . 116 , 25 . 101 , 45 , , 1 1 1 1 , 45 15 15 15 15 120 60 60 45 105 75 15 15 15 60

2 2 2 2 1 1 1 1

4 3 2 1 4 3 2 1 2 1

= = = = = = = = = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − − − − = B h B h B h B h B h B h B h B h I P P B

P P P P P P P P

• 

Observe the given matrix B and permutation matrices P1 and P2.

Numerical examples

. 6843 . 968421 . 76842 . 1 , , 1 1 1 1 , 45 15 15 15 15 120 60 60 45 105 75 15 15 15 60

1 1 1 2 1

= ≤ ≤ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − − − − =

∞ − ∞ − ∞ −

B B B I P P B

Although the given matrix IS a Nekrasov matrix, in this way we

• btained a better bound for

the norm of the inverse.

∑– Nekrasov matrices

• Given any matrix A and any nonempty proper subset S of N we

say that A is an S-Nekrasov matrix if

( )

, , S i A h a

S i ii

∈ >

( )

, , S j A h a

S j jj

∈ >

( )

( )

( )

( )

( ) ( )

. , , S j S i A h A h A h a A h a

S j S i S j jj S i ii

∈ ∈ > − −

( ) ( )

A r A h

S S 1 1

=

( ) ( )

∑ ∑

∈ + = − =

+ =

n S j i j ij jj S j i j ij S i

a a A h a A h

, 1 1 1

• If there exists a nonempty proper subset S of N such that A is an S-

Nekrasov matrix, then we say that A belongs to the class of ∑-Nekrasov matrices.

∑– Nekrasov matrices

Cvetković, Lj., Kostić, V., Rauški, S. : A new subclass of H-

• matrices. AMC (2009)

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∈ = ∈ > = = = S i for w and S i for w w w w diag W

i i n S

1 : ,..., ,

2 1

γ W

S N\S 1

Nekrasov

γ γ 1 1 . . . .

∑– Nekrasov matrices

Szulc, T., Cvetković, Lj., Nedović, M. : Scaling technique for Nekrasov matrices. AMC (2015) (in print) Cvetković, Lj., Nedović, M. : Special H-matrices and their Schur and diagonal-Schur complements. AMC (2009) Cvetković, Lj., Kostić, V., Rauški, S. : A new subclass of H-

• matrices. AMC (2009)

Nonstrict conditions

( )

. ,..., 2 , 1 , n i A r a

i ii

= ≥

DD - matrices

( )

A r a

k kk >

irreducibility for one k in N Olga Taussky (1948)

IDD-matrices

( )

. ,..., 2 , 1 , n i A r a

i ii

= ≥

( )

A r a

k kk >

for one k in N non-zero chains

• T. Szulc (1995)

CDD-matrices

Nonstrict conditions

( )

. ,..., 2 , 1 , n i A r a

i ii

= ≥

DD - matrices

( )

A r a

k kk >

irreducibility for one k in N Olga Taussky (1948)

IDD-matrices

( )

. ,..., 2 , 1 , n i A r a

i ii

= ≥

( )

A r a

k kk >

for one k in N non-zero chains

• T. Szulc (1995)

CDD-matrices

Li, W. : On Nekrasov matrices. LAA (1998)

Nonstrict conditions

S-IDD

Given an irreducible complex matrix A=[aij]nxn, if there is a nonempty proper subset S of N such that the following conditions hold, where the last inequality becomes strict for at least one pair of indices i in S and j in N\S, then A is an H-matrix.

( )

S i a A r a

i j S j ij S i ii

∈ = ≥

∑

≠ ∈

,

,

( )

( )

( )

( )

( ) ( )

. , , S j S i A r A r A r a A r a

S j S i S j jj S i ii

∈ ∈ ≥ − −

Cvetković, Lj., Kostić, V. : New criteria for identifying H-matrices. JCAM (2005)

Nonstrict conditions

S-CDD

Given a complex matrix A=[aij]nxn, if there is a nonempty proper subset S of N such that the following conditions hold, where the last inequality becomes strict for at least one pair of indices i in S and j in N\S, and for every pair of indices i in S and j in N\S for which equality holds there exists a pair of indices k in S and l in N\S for which strict inequality holds and there is a path from i to l and from j to k, then A is an H-matrix.

( )

S i a A r a

i j S j ij S i ii

∈ = ≥

∑

≠ ∈

,

,

( )

( )

( )

( )

( ) ( )

. , , S j S i A r A r A r a A r a

S j S i S j jj S i ii

∈ ∈ ≥ − −

Cvetković, Lj., Kostić, V. : New criteria for identifying H-matrices. JCAM (2005)

Cvetković, Lj., Kostić, V., Kovačević, M., Szulc, T. : Further results on H-matrices and their Schur complements. AMC (2008) Cvetković, Lj., Nedović, M. : Special H-matrices and their Schur and diagonal- Schur complements. AMC (2009) Cvetković, Lj., Nedović, M. : Eigenvalue localization refinements for the Schur

• complement. AMC (2012)

Cvetković, Lj., Nedović, M. : Diagonal scaling in eigenvalue localization for the Schur complement. PAMM (2013) Szulc, T., Cvetković, Lj., Nedović, M. : Scaling technique for Nekrasov matrices. AMC (2015) (in print) Cvetković, Lj., Kostić, V., Nedović, M. : Generalizations of Nekrasov matrices and

• applications. (2014)

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