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Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.

2 votes
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Primal identity in matrix semigroup

Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product. We call $s_1\cdots s_k$ an identity index if $M_{s_1}M_{s_ …
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139 views

Jordan Decomposition of Sparse matrix

Suppose we are given $n \times n$ rational matrix, $A$ with at most $k$ nonzero elements in each row and each column with $k \ll n$. What is the best algorithm to compute its Jordan decomposition? Wh …
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5 votes
1 answer
398 views

almost diagonal Positive semidefinite Matrix

Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices. A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in \m …
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6 votes
1 answer
541 views

On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if $x^{T}M x\geq 0$ holds for all non-negative real $x_1,x_2,\cdots,x_n$, where $x=(x_1,x_2,\cdots,x_n)^T$. Notic …
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5 votes
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On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists matri …
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