# Questions tagged [symmetric-matrices]

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

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### Quadratic form of a real symmetric matrix is bounded

If $\lambda_1>\lambda_2>...>\lambda_r$ are the different eiegenvues of a real symmetric matrix $A\in M_{n\times n}(\mathbb{R})$. $1.$ Show that the quadratic form associated to $A$ satisfies ...
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### Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
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### Triviality in the proof of the spectral theorem for real matrices which I can't get my head around [closed]

I asked the following to chatGPT: Can you prove that every symmetric real matrix has an orthonormal basis consisiting of eigenvectors? - To which it responded with: Certainly! To prove that every ...
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I am aware that for any $m \times n$ matrix $A$, we can write: Known 1: $A = U\Sigma V^T$ where $U$ is orthogonal and $m \times m$, $V$ is orthogonal and $n\times n$, and $\Sigma$ is diagonal and $m \... 2 votes 1 answer 47 views ### Rank of a matrix$A$Let $$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{13} & a_{14} & a_{24} \\ a_{13} & a_{14} & a_{24} & a_{34} \\ a_{14} & a_{24} & a_{... • 618 2 votes 2 answers 111 views ### A^3=A, {\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha implies A is symmetric? Let n\times n real matrix A satisfies {A}^{3} = A; and for any \alpha\in\Bbb R^n, we have {\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha, prove that A is symmetric. My attempt: ... • 3,593 1 vote 2 answers 50 views ### Congruent diagonalization using row and column operations Let$$A=\begin{pmatrix} 1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9 \end{pmatrix}.$$Find an invertible matrix P such that P^tAP is diagonal. Let me start by saying that I already ... • 1,482 1 vote 0 answers 35 views ### Is there such things as an inversion polynomial? In enumerative combinatorics there is a descent polynomial that counts the number of permutations in the symmetric group of size n with a descent set equal to some fixed set S. I was wondering if ... 0 votes 1 answer 27 views ### Is it true that D A P D^T = A D P D^T if P is symmetrical, positive definite and D is diagonal? I know in general, matrix multiplication is not commutative, but would it be true in this special case? D A P D^T = A D P D^T where A, D, P are all n by n matrix. But P is symmetrical and ... • 149 1 vote 0 answers 37 views ### How to decompose the Hessian matrix of a 3rd degree polynomial into 2 or more vectors/matrices? I am trying to figure out what can be said about the spectral radius of the Hessian of a 3rd degree random polynomial defined over a unit hypercube, drawing on known results in random matrix theory. ... • 91 0 votes 1 answer 37 views ### Determine if inner product over a real vector space has a certain form Verify if the following statement is true: Every inner product on \mathbb{R}^n has the form \langle v,u\rangle = v(Au), where A is a symmetric matrix with positive entries on the diagonal. I ... • 1,482 0 votes 0 answers 35 views ### Find values which make two matrices congruent For which values of a\in\mathbb{R} are the following matrices congruent?$$A=\begin{pmatrix} 1&4-a-a^2\\ 2& -1 \end{pmatrix}B=\begin{pmatrix} -a-1 & 3\\ 3 & -5 \end{pmatrix}$$... • 1,482 1 vote 0 answers 17 views ### Fastest way to divide by a symmetric positive matrix Say P_{yx} is a general (n_y,n_x) matrix. Say P_y is symmetric, positive definite, of size (n_y,n_y). I want to compute GT=P_y\backslash P_{yx} (matrix left division, or perhaps more ... • 280 0 votes 1 answer 23 views ### For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric? Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix A\in\mathbb{R}^{n\times n} with {\rm rank}(A)=r, suppose the rank decomposition of A is given as follows:$$... • 155 1 vote 1 answer 27 views ### A matrix inequality$X - \frac{1}{u^\top Xu} Xuu^\top X \succeq Y - \frac{1}{u^\top Yu} Yuu^\top Y$Problem. Let$n\ge 2$. Let$X, Y$be$n\times n$real symmetric positive definite (PD) matrices with$X \succeq Y$(i.e.$X - Y$is positive semi-definite (PSD)). Let$u\ne 0$be a$n\times 1$real ... • 40.3k 2 votes 0 answers 26 views ### For a symmetric matrix$B$and following four relevant matrices$P,Q,C,D$, what's the relation between$QP$and$CDC^{\rm T}$? Suppose$B\in\mathbb{R}^{n\times n}$is a symmetric matrix with${\rm rank}(B)=r$. Then$B$is equivalent to$\tilde{B}$in (1), where$I_{r}$denotes the identity matrix of order$r$. That is, there ... • 155 1 vote 1 answer 56 views ### If$B$is PD in null space of$A\in \mathbb{R}^{m\times n}$then exists$r \in \mathbb{R}$such that$B+rA^TA$is PD If a symmetric matrix$B$is PD in null space of$A\in \mathbb{R}^{m\times n}$, with rank$A=m<n$, then exists$r \in \mathbb{R}$such that$B+rA^TA$is PD. What I've tried is the following: If$x\...
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Let $A , B \in M_{4 \times 4}(\mathbb{R}_{\geq 0})$. Matrices $A, B$ are $\textbf{permutation similar}$ if there exists a permutation matrix such that $A= PBP^T$. Define $\mathrm{diag}(A)$ to be the ...