Questions tagged [symmetric-matrices]

Question 1

I observed that for matrices of following forms, their product are also symmetric: a b c b c b c b a or perhaps also other similar forms. Truth to be told, I'm ...

Question 2

For $i=1,2$, let $B_i\in\mathrm{SL}_n(\mathbb{Z})$ be two symmetric positive definite matrices. We define their automorphism groups as $$\mathrm{Aut}(B_i)=\{g\in\mathrm{GL}_n(\mathbb{Z})\mid\ gB_ig^{\...

Question 3

Let $A,B,C \in \mathbb{R}^{d \times d}$ be symmetric positive semi-definite with $A$ strictly positive definite. Is it true that $$||BM^{-1}||_2 \le ||BN^{-1}||_2, \quad M := A + B + C, \quad N := A + ...

Question 4

Exactly, I can't understand the real symmetric matrix is diagonalizable only from the symmetry. I can prove that the diagonalization of this kind of matrices by mathematical induction,as in Artin's ...

Question 5

Let the $n \times n$ real matrices ${\bf A}, {\bf B}$ be symmetric and positive semidefinite. If ${\bf A} \succeq {\bf B}$, can one conclude that ${\bf A}^{\frac12} \succeq {\bf B}^{\frac12}$, i.e., ...

Question 6

Assume that $A$ is an SPD matrix and let $\langle\cdot,\cdot\rangle$ be an inner product on a vector space $X$. Can you, in general, obtain an expression $$ \langle Ax, y \rangle = C\langle x,y\rangle ...

Question 7

I am trying to implement a loss function from a paper which is: $$\mathscr{L}_{\text{GL}} = \sum_{i, j=1}^n \lVert x_i- x_j \rVert_2^2S_{ij} + \gamma \lVert S\rVert_F^2$$ where $x\in \mathbb{R}^n$ is ...

Question 8

I don't know if the title is clear, but basically: If $A$ is an $n \times n$ real, symmetric matrix, is it true that $\langle Ax, y \rangle = \langle x , Ay\rangle$ for any inner product $\langle \...

Question 9

Let $A$ be a real symmetric matrix of size $n$. Consider the matrix-equation $$ X^5 + 5X = A. $$ Prove the following claims: (1) This equation has a solution which is real and symmetric and is a ...

Question 10

Let $A := \operatorname*{diag}(a_{1},\dots,a_{n})$ and $B := \operatorname*{diag}(b_{1},\dots,b_{m})$ be diagonal real matrices. Let $$ M := \begin{pmatrix} A & J \\ J^{\top} & B \end{pmatrix} ...

Question 11

Suppose that I have two matrices $A$ and $B$ which are both symmetric: $A=A^T, B=B^T$. Moreover, I know how to diagonalize both $A$ and $B$. Now I would like to define $T=A^{1/2}BA^{1/2}$, which is ...

Question 12

P. Shiva Shankar — High school student I recently observed an interesting and seemingly undocumented property of matrix multiplication: Let A be a 3×3 symmetric matrix in which each row is an ...

Question 13

Question: $A$ is a $3 \times 3$ symmetric matrix, six of whose entries are $1$ and three entries are $0$, then the number of distinct values of $\det(A)$ is - My try: Case $1$: the diagonal has all ...

Question 14

I am new to lie theory and representation theory. I heard about this interesting factorization known as the Bipolar decomposition which uses the Mostow decomposition. The article is https://www....

Question 15

I am reading Multivariable Mathematics by Theodore Shifrin. From page 215, Let $A$ be a positive definite matrix. Then, $$a_{11}=\left[ \begin{array}{cccc} 1 & 0 & \cdots & 0 \end{array} \...

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Questions tagged [symmetric-matrices]

This seem to be a sufficient condition for non-commutative matrix product to be symmetric. Is it also necessary? [closed]

Automorphisms of integral quadratic forms

Is $||B(A + B + C)^{-1}|| \le ||B(A+B)^{-1}||$ for positive semi-definite matrices?

Why is a real symmetric matrix diagonalizable?

Does the square root preserve the Löwner order? [duplicate]

Pull SPD matrix out of inner product

Converting item-wise summation of difference between vectors to calculation using trace

In Euclidean space, are symmetric operators self adjoint?

Existence and uniqueness of real symmetric solutions of $X^5 + 5X = A$

Eigenvalues of $M=\begin{pmatrix} A & J \\ J^{\top} & B \end{pmatrix}$ with $A,B$ diagonal

Efficiently computing the trace of products of diagonalizable matrices

Do symmetric 3×3 matrices with rows in arithmetic progression always produce A.P. columns in matrix products? [closed]

Finding the number of distinct values a certain symmetrical (0,1) determinant can take

Geometric interpretation of the matrix exponential of imaginary symmetric and imaginary skew symmetric matrices?

Theodore Shifrin's Multivariable Mathematics — How to prove that symmetric matrix $B$ is positive definite?

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