[Paper Review] Product of positive semi-definite matrices

Jianlian Cui, Chi-Kwong Li+1 more|arXiv (Cornell University)|Jun 30, 2015

graph theory and CDMA systems3 citations

TL;DR

This paper provides a complete characterization of complex square matrices that can be expressed as the product of three or four positive semi-definite matrices, filling a critical gap in the theory of matrix factorizations. Using unitary similarity transformations to reduce matrices to upper triangular form with invertible and nilpotent blocks, the authors derive necessary and sufficient conditions based on the structure of the blocks and the numerical range, enabling an algorithmic determination of the minimal number of positive semi-definite matrices needed for any given matrix with nonnegative determinant.

ABSTRACT

It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of $k$ positive semi-definite matrices but not fewer, for $k = 1,2,3,4,5$.

Motivation & Objective

• To close the gap in characterizing matrices requiring exactly three or four positive semi-definite matrices in their product, as prior work only covered one, two, and five factors.
• To provide a complete and algorithmic criterion for determining the minimal number of positive semi-definite matrices needed to express any complex square matrix with nonnegative determinant.
• To resolve the open question of whether a matrix in upper triangular form with invertible and nilpotent blocks can be written as a product of three positive semi-definite matrices based on the properties of its diagonal block.
• To establish a necessary and sufficient condition for an invertible matrix to be expressible as a product of three positive definite matrices, using the numerical range and eigenvalue arguments.

Proposed method

• Transform any matrix to upper triangular form via unitary similarity, separating it into an invertible block $ T_1 $ and a nilpotent block $ T_2 $, with off-diagonal block $ R $.
• Use the numerical range $ W(A) $ as a key tool to analyze the spectral properties of the invertible block $ T_1 $, particularly whether 0 lies in its interior.
• Establish equivalence conditions for a matrix to be a product of two positive semi-definite matrices using unitary invariance and similarity transformations.
• Prove that a matrix $ T $ is a product of three positive semi-definite matrices if and only if either $ R \neq 0 $ or $ T_2 \neq 0 $, or both $ R = 0 $, $ T_2 = 0 $, and $ T_1 $ is a product of three positive definite matrices.
• Provide a criterion for $ T_1 $ to be a product of three positive definite matrices: either $ \det(T_1) > 0 $ and $ 0 \in \mathrm{int}(W(T_1)) $, or $ 0 \notin \mathrm{int}(W(T_1)) $ but $ W(T_1) $ contains a positive number and the sum of eigenvalue arguments is zero.
• Construct an algorithm that determines the minimal $ k \in \{1,2,3,4,5\} $ such that a matrix $ A \in M_n $ with $ \det(A) \geq 0 $ can be written as a product of $ k $ positive semi-definite matrices, based on the structure of its unitary similarity form.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a complex square matrix to be expressible as the product of exactly three positive semi-definite matrices?
• RQ2 Can the characterization of matrices requiring four positive semi-definite matrices be completed, especially in cases where the matrix is not a scalar multiple of the identity with negative real scalar?
• RQ3 Under what conditions is an invertible matrix $ T_1 $ the product of three positive definite matrices, particularly when the numerical range of $ T_1 $ does not contain zero in its interior?
• RQ4 Is the property of being a product of three positive semi-definite matrices preserved under unitary similarity when the matrix is in upper triangular form with an invertible diagonal block and a nilpotent block?
• RQ5 What is the algorithmic procedure to determine the minimal number $ k \in \{1,2,3,4,5\} $ for which a given matrix with nonnegative determinant can be expressed as a product of $ k $ positive semi-definite matrices?

Key findings

• A matrix $ T = \begin{bmatrix} T_1 & R \\ 0 & T_2 \end{bmatrix} $ with $ T_1 $ invertible and $ T_2 $ nilpotent is a product of three positive semi-definite matrices if and only if either $ R \neq 0 $, $ T_2 \neq 0 $, or both $ R = 0 $, $ T_2 = 0 $, and $ T_1 $ is a product of three positive definite matrices.
• An invertible matrix $ T_1 $ is a product of three positive definite matrices if and only if either $ \det(T_1) > 0 $ and $ 0 \in \mathrm{int}(W(T_1)) $, or $ 0 \notin \mathrm{int}(W(T_1)) $ but $ W(T_1) $ contains a positive number and the sum of the arguments of its eigenvalues is zero.
• If a matrix $ A $ is a scalar multiple of the identity with a negative real scalar, i.e., $ A = \alpha I_n $, $ \alpha \notin [0, \infty) $, then it requires exactly five positive semi-definite matrices and cannot be expressed with fewer.
• The algorithm correctly determines the minimal $ k \in \{1,2,3,4,5\} $ for any matrix $ A \in M_n $ with $ \det(A) \geq 0 $, based on its unitary similarity form and the properties of its blocks and numerical range.
• The paper resolves a conjecture by showing that the condition $ \det(T_1) > 0 $ is necessary but not sufficient for $ T $ to be a product of three positive semi-definite matrices, and provides a full characterization via the numerical range and eigenvalue argument sum.
• The characterization of matrices requiring exactly four positive semi-definite matrices is complete: such matrices are those that are not products of fewer than four, specifically those of the form $ T_1 \oplus 0_p $ where $ T_1 $ fails both conditions (i) and (ii) for being a product of three positive definite matrices.