# Questions tagged [matrices]

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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### Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
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### Do there exist graphs whose adjacency matrix is positive semi-definite? [closed]

If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
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### “euler form” of a unitary matrix? [closed]

A little background: I'm a student of physics (forgive me for a difference in language or the lack of familiarity with this topic), trying to solve an equation of the form: \begin{equation} \mathcal{...
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### trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form? Such an expression would naturally generalise some invariant random matrix ensembles. ...
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### Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
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### A linear algebra question regarding the eigenvalues of the product of a diagonal matrix and a projection matrix

I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few ...
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Def.: A non-negative $n \times n$ matrix $A$ is called a non-primitive if there is no an integer $k$ such that all entries of $A^k$ are positive. Def.: Let ${\bf A}=(a_{i,j})$ and ${\... 0answers 34 views ### Minimum rank of a product of two block diagonal matrices with an arbitrary matrix Let us assume that we have an arbitrary full-rank$l\cdot b \times l\cdot p$matrix,$\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an$m \times ...
I've got an $n\times n$ matrix that is upper-triangular (all zeros below diagonal), diagonal entries are positive, and entries that are not on the diagonal are either $0$ or $1$. An example matrix ...
Let $\mathbf{F}_q$ be a finite field of order $q$ ($q$, an odd prime, or a power of the same). I know that for the matrix algebra $M(n,\mathbf{F}_q)$ (or $gl(n,q)$), the maximal dimension for a ...