Let I be the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> identity matrix and A be any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is a well-known fact that the block matrix <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\begin{bmatrix} I & A^*\\ A & I \end{bmatrix}$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="]" open="["> <mrow> <mtable> <mtr> <mtd> <mi>I</mi> </mtd> <mtd> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>A</mi> </mrow> </mtd> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </InlineEquation> is positive semidefinite if and only if A is contractive. We aim to generalize this to higher dimensional block matrices. Two possible generalizations are discussed and one of them is fully characterized in terms of singular values of A,  while the other’s positivity properties are discussed. To obtain deeper results and conclude the analysis about more block matrices, a variety of new techniques have been introduced.

Positivity of block matrices

期刊论文
发表日期：

2026年3月3日

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Anju Rani,
Yogesh Kapil,
Mandeep Singh,
Rajinder Pal

摘要

Let I be the $n\times n$ $n \times n$ identity matrix and A be any $n\times n$ $n \times n$ matrix over $\mathbb {C}.$ $C .$ It is a well-known fact that the block matrix $\begin{bmatrix} I & A^*\\ A & I \end{bmatrix}$ $[\begin{matrix} I & A^{*} \\ A & I \end{matrix}]$ is positive semidefinite if and only if A is contractive. We aim to generalize this to higher dimensional block matrices. Two possible generalizations are discussed and one of them is fully characterized in terms of singular values of A, while the other’s positivity properties are discussed. To obtain deeper results and conclude the analysis about more block matrices, a variety of new techniques have been introduced.