# Tagged Questions

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

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### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
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### Why does the objectivity rule out the convexity?

In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...
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### Matrix transformation [closed]

I want to show that $(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$ is equivalent to $M_\gamma$ has no purely imaginary ...
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### Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory). Preferably the ...
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### The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
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### inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
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### Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...
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### How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?
Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...