# 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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### Is the SVD optimal (by Eckart Young theorem) because maximal data variance is captured from both the column and row space?

I am asking this question seeking validation of an intuitive understanding of the veracity of the Eckart-Young theorem which struck me in my study of the SVD and Principle Component Analysis. The ...
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### An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality. Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
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### Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?

Let $A$ be a matrix whose entries are given by a polynomial, $$A_{ij} = p(\lambda_i, \lambda_j)$$ where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric. Are there standard methods ...
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### Spectral abscissa of symmetric matrix with skew-symmetric perturbation

I am interested in bounds on the minimal distance between the spectral abscissa $\max_{\lambda\in\sigma(A)}\mathrm{Re}\lambda$ of a matrix $A$ and the eigenvalues of its perturbated version $A+S$. In ...
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### Minimum distance of a symmetric matrix to diagonal matrices

Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix and $n\geq 2$. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for ...
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### Comparison of two similarity matrices

English is not my first language, so please excuse any mistakes. I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...
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### A unitary matrix of functions [closed]

If $A(z)=[A_{ij} (z)]$ is an $n\times n$ unitary matrix valued functions. Is there a characterization of such matrix if: (1) the entries are analytic functions on a set $D$. and (2) if the ...
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### Operator norm of square root of matrix vs original

If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...
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Let $X\in M_2(\mathbb{C})$ s.t. $W(X)\subseteq\text{Conv}\{\lambda_1,\cdots,\lambda_k\}$ and $\Vert X\Vert\leq\text{max}_{1\leq j\leq k}\vert\lambda_j\vert$. Then does it true that there exist $H_1, ... 0answers 101 views ### Distance between two algebraic sets We are in$M_n(\mathbb{R})$equipped with the Frobenius norm$||A||^2=tr(AA^T)$. Let$Z=\{(A,B)\in M_n(\mathbb{R})^2;A^2-AB-B^2=0\}$and$T=O(n)^2$. It is easy to see that$Z\cap T=\emptyset$and ... 2answers 157 views ### Commutant of the conjugations by unitary matrices Let$\mathcal{L}(\mathbb{C}^{n \times n})$denote the algebra of all linear mappings from$\mathbb{C}^{n \times n}$to$\mathbb{C}^{n \times n}$and let$\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true? If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then \$W(B\otimes (X+I_n))\subseteq W(B\...