Questions tagged [symmetric-matrices]

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

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $A^T=A$.

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB = BA$. So for integer $n$, $A^n$ is symmetric if $A$ is symmetric. If $A^{−1}$ exists, it is symmetric if and only if $A$ is symmetric.

The complex generalization is a hermitian matrix, a square matrix equal to its conjugate transpose. This is often denoted $A=A^{H}$ or $A=\overline{A^T}$; see for more information.

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Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix

If $M$ is a symmetric positive-definite matrix, is it possible to get a positive lower bound on the smallest eigenvalue of $M$ in terms of a matrix norm of $M$ or elements of $M$? E.g., I want $$\lambda_{\text{min}} \geq f(\lVert M \rVert)$$ or…
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Is there any intuition why the following matrix is positive semidefinite?

I have the following 8 by 8 square matrix, which is positive semidefinite: \begin{bmatrix}3&1&1&-1&1&-1&-1&-3\\1&3&-1&1&-1&1&-3&-1& \\ 1&-1&3&1&-1&-3&1&-1 \\ -1&1&1&3&-3&-1&-1&1 \\ 1&-1&-1&-3&3&1&1&-1 \\ -1&1&-3&-1&1&3&-1&1 \\ -1&-3&1&-1&1&-1&3&1 …
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Over which fields (besides $\mathbb{R}$) is every symmetric matrix potentially diagonalizable?

Over which fields (besides the well-known $\mathbb{R}$) is every symmetric matrix potentially diagonalizable? A matrix is potentially diagonalizable in a field $F$ if it is diagonalizable in the algebraic closure of $F$. It appears to me that the…
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Determining if a symmetric matrix is positive definite

I have a symmetric matrix where all non-diagonal elements are positive and identical, and all diagonal elements are identical as well. For example, the $3 \times 3$ version of this matrix has the following form: $$ \left( \begin{array}{ccc} 2a+b &…
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Why is the maximum Rayleigh quotient equal to the maximum eigenvalue?

(Note: I'm only interested in real-valued matrices here, so I'm using "transpose" and "symmetric" instead of the more general "transjugate" and "Hermitian" in the hope that it will simplify the proof. But the theorem apparently holds for…
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Inverse of a symmetric positive definite matrix

If a matrix is symmetric and positive definite, determine if it is invertible and if its inverse matrix is symmetric and positive definite. I know that "if a matrix is symmetric and positive definite, then its inverse matrix is also positive…
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How to prove that $A^2$ is a symmetric matrix?

Conjecture 1 : Let $A$ be a real matrix such that $A^5=A A^T A A^T A$. Then $A^2$ is a symmetric matrix. (here $A^T$ denotes the transpose of a matrix A). I guess that the following is also true : Conjecture 2 : If $A^{2n+1}=AA^TAA^T\cdots…
math110
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Did I just discover a new way to calculate the signature of a matrix?

Due to the complains for more clarity down below I've cut my post into segments. Feel free to skip right to Definitions, Algorithm & Conjecture. If this is not clear enough, then I'm afraid I can't help it. Story I'm taking a course on linear…
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Maximizing trace of mixed products of two real symmetric matrices

Let $A$, $B$ be two $N \times N$ real symmetric matrices whose entries i.i.d.r.v. from a mean 0, variance 1 distribution. Let $I, J$ be even positive integers, and let $i_k, j_k$ for $k = 1,\ldots,n$ be arbitrary finite sequences of positive…
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Is there a formula for the expansion coefficients of powers of an inner product?

I would like to expand the following expression $$\left(\sum_{i,j=1}^N \,x_i A_{ij} x_j\right)^n$$ where $\mathbf A$ is a symmetric $N\times N$ matrix, $\mathbf {x}$ is an $N$-component vector, and $n$ is a non-negative integer power. The expansion…
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Notation for the set of symmetric matrices and symmetric positive definite matrices

I would like to know if there exists a notation for the set of symmetric matrices and symmetric positive definite matrices. For instance, the set of $N \times N$ matrices with real entries is denoted as $\mathbb{R}^{N \times N}$.
John Smith
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Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 & & -1& & \ddots& \\ -1& & \ddots& &\ddots & & -1 \\ & -1& & & &-1& …
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Can the product of two nonsymmetric matrices be symmetric?

I was wondering if the product of two nonsymmetric matrices can ever be a symmetric matrix. Honestly I would not know how to tackle this problem.
user168764
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How to find a symmetric matrix that transforms one ellipsoid to another?

Given two origin-centered ellipsoids $E_0$ and $E_1$ in $\mathbb{R}^n$, I'd like to find an SPD (symmetric positive definite) transformation matrix $M$ that transforms $E_0$ into $E_1$. Let's say $E_0$ and $E_1$ are specified by SPD matrices that…
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Why do positive definite symmetric matrices have the same singular values as eigenvalues?

I realize that this is because when the eigenvalues are either 0 or 1 they will have the same square root. But why does this happen?