# Questions tagged [symmetric-matrices]

A symmetric matrix is a square matrix that is equal to its transpose.

1,302 questions
Filter by
Sorted by
Tagged with
2answers
39 views

### Is a symmetric real matrix similar to a diagonal matrix through an orthogonal matrix?

Definition Two matrices $A$ and $B$ are said similar if there exist an inverible matrix $P$ such that $$B=PAP^{-1}$$ Definition A square matrix $A$ is said orthogonal if it is invertible and its ...
0answers
10 views

### Ordered eigenvalues of circulant matrix when using DFT

Let $\mathbf{C} \in \mathbb{R}^{n \times n}$ be a circulant matrix: \begin{equation} \mathbf{C} = \begin{pmatrix} {c_0} & {c_1} & {\dots} & {c_{n-2}} & {c_{n-1}} \\ {c_{n-...
1answer
40 views

### eigenvectors of a real symmetric matrix are always orthogonal

As we know by the famous theorem "eigenvectors corresponding to distinct eigenvalues are orthogonal for a real symmetric matrix" can this result be also true for the same eigenvalues My ...
1answer
57 views

### Given the distance measure for two points and the point $p = (0,2)$, which of the following points have the same distance as $p$ from the origin?

Given the distance measure dist: $$\mathrm{dist}(x,y) = \sqrt{(x_1-y_1,x_2-y_2)\begin{pmatrix} 3 & 0\\ 0 & 4 \end{pmatrix} (x_1-y_1,x_2-y_2)^{T} }$$ for two dimensional points and the ...
0answers
29 views

1answer
67 views

### If $S$ is symmetric positive definite and $SA$ symmetric, is then $A$ symmetric?

We are given real matrices $S$ and $A$. We know that $S$ is symmetric positive definite and that $SA$ is symmetric. Is A necessarily symmetric then? I've figured out that if $A$ is symmetric, then $S$ ...
0answers
45 views

### Invariants of a symmetric bilinear form

The following is a theorem from the Linear Algebra Textbook by Friedberg, Insel, and Spence (5th Edition). Theorem 6.38 (Sylvester's Law of Inertia). Let $H$ be a symmetric bilinear form on a finite-...
0answers
73 views

0answers
24 views

### Show proofs for inverse of these singular matrix.

I tried really hard, but I have no idea how to approach this question. A and B matrix are not invertible, so inverse does not exist. So, how do I go about proving them ? Simply saying they do not have ...
3answers
69 views

0answers
55 views

### Why is the derivative of $\frac{\partial{Tr(AXB)}}{\partial{X}}$ for symmetric matrix X equal to the asymmetric result?

I'm a bit confused of the fact that an online matrix calculus calculator (https://www.matrixcalculus.org/matrixCalculus) gives the same result for the derivative w.r.t. a symmetric matrix $X$ and a ...
1answer
60 views

### Inequality between symmetric matrices

Let $N$ be symmetric positive definite with $N-S$ positive semidefinite and let $Y := N (2N -S)^{-1} N$. Then $\frac{1}{2} v^{\top} N v \leq v^{\top} Y v$. How does this follow?
1answer
21 views

### Bounds on extremal eigenvalues of gram matrix with diagonal entries in $[a, b]$ and off-diagonal entries in $[c, d]$

Consider a square $n \times n$ matrix $H = A^T A$ where $A$ is an $m \times n$ with $m \ge n$. Knowing that $a \le H_{ii} \le b$ for all $i = 1, \ldots, n$ and that $c \le H_{ij} \le d$ for $i \neq j$,...
0answers
40 views

### On the Symmetrized Product $AB+BA$

I have encountered the matrix $P=P^T$, \begin{align*} P:=AB+BA, \end{align*} where $A,B\in\mathbb{R}^{n\times n}$ and (in my case) $A=A^T$, $B=B^T$ and $1 \succeq A,B\succeq 0$. The matrix $P$ is ...
0answers
20 views

1answer
37 views

### About real symmetric matrix multiplied by diagonal matrix [closed]

Recently, I found an important matrix in analog circuit domain and it need to be proved diagonalized. Then I try to resolve it into a small problem that is: if there are a $n\times n$ real symmetric ...
2answers
45 views

1answer
63 views

### Spectral radius and positive definite matrix

Let $A$ be a $m \times m$ matrix. Show that there exists a $m \times m$ positive definite matrix $B$ such that $B - A^H B A \succ 0$ if the spectral radius $\rho(A) < 1$ Let $B= B^{1/2}B^{1/2}$, ...
0answers
46 views

### Measurability of a solution to $Ax=b$ for measurable symmetric matrix

Let $(\Omega, \mathcal F)$ be a measurable space; $A:\Omega \to \mathbb R^{n\times n}$ be a measurable matrix-valued function and $b:\Omega\to \mathbb R^n$ be a measurable vector-valued function. ...
2answers
95 views

### If $A<B$, does it follow that $A^2\leq B^2$?

Suppose $A$ and $B$ are positive semidefinite matrices satisfying $A\leq B$ (meaning that $x^TAx \leq x^TBx$ for any vector $x$). Does it follow that $A^2\leq B^2$? It certainly follows if $A$ and \$...