# Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

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### If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
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### Does every positive semidefinite hankel matrix obeys one Vandermonde decomposition?

I'm reading the paper(I can't find one arXiv version of this paper...) and suspect the correctness of one theorem inside. A hankel matrix $H$ is a square matrix in which each ascending skew-diagonal ...
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### How to prove this variation-of-constants formula?

If $A ,B\in \mathbb{R}^{n\times n},v\in\mathbb{R}^n$,then from the variation-of-constants formula $$\exp((A+B)\tau)v=\exp(A\tau)v+\int_0^\tau \exp(As)B\exp((A+B)(\tau-s))vds$$. In my opinion,this ...
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### Relations between row and columns of orthogonal matrix

Given a orthogonal matrix $Q \in \mathbb{R}^{n\times n}$, we know $Q^{\top}$ is also orthogonal. Let $Q$ represent a linear transformation from an Euclidean space to itself, then reading from the ...
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### When does $A_1 A_2 A_3\ldots \xrightarrow{a.s}0$ for IID random matrices $A_i$?
Suppose $A_1,A_2,A_3,\ldots$ is an infinite sequence of $d\times d$ matrices sampled IID from some distribution. Under which conditions does the product converge to zero almost surely? The hard case ...