A matrix pair (A, B) of Hermitian matrices is definite if there exists some real linear combination of the matrices A and B which is a positive definite matrix. Determining whether a pair of matrices is definite is a nontrivial problem, particularly challenging when dealing with large matrices. In this paper, we propose a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-basic subspace algorithm and two specialized versions for detecting the definiteness or indefiniteness of a large matrix pair <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${(A,B)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The proposed subspace algorithms are based on the iterative formation of small reduced Hermitian matrix pairs and the detection of their definiteness or indefiniteness. In particular, if a reduced pair is not definite, then the original pair is also not definite. The proposed algorithms are applicable to large-scale Hermitian matrix pairs <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${(A,B)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, even when the matrices are too large to fit entirely in memory and are given implicitly through procedures <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x\mapsto Ax$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <mi>A</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x\mapsto Bx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <mi>B</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> for a given vector x. The numerical performance of the proposed algorithms is demonstrated through the experiments.

Detecting large definite Hermitian matrix pairs by $\theta $-subspace algorithms

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发表日期：

2025年9月10日

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Marija Miloloža Pandur,
Ivana Kuzmanović Ivičić

摘要

A matrix pair (A, B) of Hermitian matrices is definite if there exists some real linear combination of the matrices A and B which is a positive definite matrix. Determining whether a pair of matrices is definite is a nontrivial problem, particularly challenging when dealing with large matrices. In this paper, we propose a $\theta $ $θ$ -basic subspace algorithm and two specialized versions for detecting the definiteness or indefiniteness of a large matrix pair ${(A,B)}$ $(A, B)$ . The proposed subspace algorithms are based on the iterative formation of small reduced Hermitian matrix pairs and the detection of their definiteness or indefiniteness. In particular, if a reduced pair is not definite, then the original pair is also not definite. The proposed algorithms are applicable to large-scale Hermitian matrix pairs ${(A,B)}$ $(A, B)$ , even when the matrices are too large to fit entirely in memory and are given implicitly through procedures $x\mapsto Ax$ $x \mapsto A x$ and $x\mapsto Bx$ $x \mapsto B x$ for a given vector x. The numerical performance of the proposed algorithms is demonstrated through the experiments.

Detecting large definite Hermitian matrix pairs by \(\theta \)-subspace algorithms

摘要