# Questions tagged [matrix-analysis]

The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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### When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?

By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that $$C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right]$$ where $e_1,\dots,e_n$ are the standard ...
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### Sum of holomorphic squares?

Consider a variable $z \in \mathbb{R}^n$ and assume $u(z) \in \mathbb{R}^m$ and $H(z) \in \mathbb{R}^{m \times m}$. Further assume that $H(z)$ is symmetric positive definite for every $z$. Consider ...
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### The eigenvectors of adding a particular rank one matrix to the circulant matrix

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$. Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
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### Diagonalization of symmetric matrices of functions

I asked this question some time ago in MSE but I didn't recieved any feedback. https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions This problem arised ...
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### Maximizing the norm of a sum of Hermitian matrices

Consider the following problem: Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
1 vote
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### On a matrix trace inequality

Let $\mathcal H(n)$ be the set of $n\times n$ Hermitian matrices, and $\mathcal S(n) \subset \mathcal H(n)$ be the subset of density matrices, i.e., $A \in \mathcal S(n)$ iff $A$ is Hermitian, ...
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### Can the Jordan decomposition of a matrix be computed in a backwards stable way?

Let $PJP^{-1}$ denote the Jordan decomposition of $M$. The matrix $J$ is a direct sum of Jordan blocks; it is unique up to permutation of the Jordan blocks. The matrix $P$ is not unique. There are two ...
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### Diagonal entries and determinant of positive definite matrices

Given a $3x3$ symmetric, positive matrix $\lambda |x|^2 \leq x^TAx \leq \Lambda |x|^2$, let us denote $a_{11}$, $a_{22}$ and $a_{33}$ to be the diagonal elements. Furthermore, let us denote the values ...
1 vote
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### A question about the sign of quadratic forms on nonnegative vectors

Let $M$ be a real square matrix of order $n\ge 3$. Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$. Can ...
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### One question on circulant $(-1,1)$-matrices

Let $n > 13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property: $$AA^T=(n-1)I+J$$ where $I$ is the $n\times n$ identity matrix and $J$ ...
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### One question on block-circulant matrices

Circulant matrices are very useful in digital image processing. I found the general formula for determinant of circulant matrix. But I think it is not suitable for block-circulant matrices. For ...
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### The upper bound for the eigenvalues of a real symmetric matrix

Let $A$ be a symmetric matrix in $M_n(\mathbb{R})$. Except Gershgorin circle theorem, is there any other theorem to say us about the upper bound of the eigenvalues of $A$?
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### $\mathbb{P}(\|A^Tx\| \ge \epsilon \|A\|\|x\|) \ge \delta$ for all $x \in\mathbb{R}^n$

Let $r$ and $n$ be integers such that $1 \le r \ll n$, and $\|\cdot\|$ denote the Euclidean norm of vectors or the spectral norm of matrices. Suppose that $\mathcal{D}$ is a probability distribution ...
For a given natural number $n$, let us consider $E_n=\{0\leq k\leq n-1 : k\equiv_41 \}$. Suppose that $E_n$ including $k_1<k_2\cdots < k_m$. Consider the following matrix $A$: A=\left(\cos\...
Let us consider the matrix $C=A_1+A_2$ where : $A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$ $A_2$ is the the $n$ by $n$ block ...