# 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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### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
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### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
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### An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
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### Two questions around the $abc$-conjecture

Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers. The abc-conjecture can be formulated using these two metrics as: For ...
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### A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
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### SVD-type decomposition for the tensor product of three Hilbert spaces?

(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
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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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### An $L^{\infty}$ version of principal component analysis?

I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal. I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
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### Spectral norm bound on smooth primary matrix function perturbation

Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...
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### Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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### Bound on difference of log of unitary matrices

Suppose I have two unitary matrices $u, v$ such that $\|u-v\|<\epsilon$ in the operator norm. Is there a way to bound the quantity $\|\log u-\log v\|$? We can assume that $\epsilon$ is sufficiently ...
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### Product of values of a matrix-valued function over $S^1$

Assume you have a function $f:S^1 \rightarrow \mathrm{GL}_d(\mathbb{C})$ whose coefficients are Laurent polynomials $f_{i,j}(q)\in \mathbb{C}[q^{\pm 1}].$ I am interested in getting conditions for ...
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### Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
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### Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $\{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

I am not sure whether this question fits this forum (I will delete it if not appropriate here). But I asked this on MSE over a week ago with no answer and then put a bounty still got no answer. Here ...
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### Maximizing a certain eigenvalue ratio

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
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### A minimal eigenvalue inequality

Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two ...