# 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 there an explicit formula for the hessian of “Determinant”?

Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
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### Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$. The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...
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Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ A\|\le\frac{\... 2answers 155 views ### Commutant of the conjugations by unitary matrices Let \mathcal{L}(\mathbb{C}^{n \times n}) denote the algebra of all linear mappings from \mathbb{C}^{n \times n} to \mathbb{C}^{n \times n} and let \mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{... 1answer 350 views ### Equivalent metrics on symmetric positive definite matrices By similar arguments as for the proof of the golden-thompson inequality (see "Log majorization and complementary Golden-Thompson type inequalities" by T.Ando and F.Hiai) we can show that for all A,B ... 2answers 677 views ### How to solve a non-homogeneous quadratic matrix equation? I am looking to solve the following matrix equation for G$$GHG + M = 0$$where G, H, and M are square, symmetric, real matrices. H is negative-definite and M is positive-definite. G ... 2answers 257 views ### The structure of the n-th power of a special matrix Assume the following matrix$$ C_p^{(a,b)}:=\left( \begin{array}{cccccc} a &a &0 &\cdots &\cdots &0 \\ 0 &0 &a &\ddots &\ddots &\vdots \\ \vdots &\ddots &...
Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector? By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$. P.S. This is ...
Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows \begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} \...