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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.

1854 questions
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On powers of symmetric matrices.

What is the best way to show if $A$ is symmetric then $A^2$ is as well using eigenvalues?
Turbo
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Bounds for the rank of a symmetric matrix

Let $A$ be an $n \times n$ real symmetric matrix. What can be said about lower bounds for the rank of $A$ when the off diagonal elements are small relative to those on the diagonal? Question: In particular, suppose the diagonal elements of $A$ are…
user2052
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How to prove if a nxn matrix A is symmetric

If A is an nxn matrix such that rank(A) = 1 and the null space of A is the orthogonal complement of its column space, then how do I show that A is a symmetric matrix? I have proven that the null space of A and A-transpose are equal. How do I proceed…
Ritik Kumar
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Sum of product of positive matrices

Let $A,B$ be two positive hermitian matrices in $M_n(\mathbb{C})$. Is $AB+BA$ a positive hermitian matrix ? This is almost the same question as Question, but it seems to me the response was not entirely complete. Is there a classical example of a…
Chr
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Find a real symmetric matrix $A$

Is there a real symmetric matrix $A$ satisfying the following two conditions? $A$ is not orthogonal. There is a positive integer $m >1$, such that $$A^m=I.$$
yao4015
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When does a product of two symmetric matrices commute?

I have two $n\times n$ symmetric matrices. I know that $\operatorname{tr}(AB)=\operatorname{tr}(BA)$. But it does not mean $AB=BA$. I wonder what is the condition for $AB=BA$.
Simon Lee
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Generating symmetric pascal matrix

Symmetric pascal matrix defined as $S=\binom{i+j}{i}$ , show that if $i=j$ by matrix multiplication we have $$S^3=\sum_{k_1 , k_2 =0}^{q^\alpha-1} \binom{i+k_1}{i}\binom{k_1 +k_2}{k_1}\binom{k_2 + j}{k_2}=q^2\varepsilon \binom{i+j}{i} +1$$ where $q$…
Andrea
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Does $A^{\star} A$ and $A A^{\star}$ always have same eigenvalues except one has zero as an eigenvalue?

I wonder this because for a $m \times n$ matrix $A$, we have that $A^{\star}A$ and $A A^{\star}$ have the same $\textbf{non-zero}$ eigenvalues. Would this mean that one of them (the one with a higher dimension) ALWAYS has zero as an eigenvalue…
Morris C.
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If $A$ is real symmetric matrix

If $A$ is real symmetric matrix then a)does not contain $0$ eigenvalue b)at least one eignvalue positive. pick correct statement 1)option a is correct 2)option b is correct 3)both option a and b is correct
Halima.Khatun
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Non positive-definiteness of some binomial matrices

Experience shows that the following matrix $$ A_4 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 3 & 4 & 0 \\ 1 & 3 & 6 & 0 & 0 \\ 1 & 4 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{pmatrix}, $$ which could be called a binomial matrix, or a Pascal…
André Bellaïche
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If matrix $A$ is not symmetric matrix then $A^{-1}$ is not symmetric.

As I mentioned before the question is If matrix $A$ is not symmetric matrix then $A^{-1}$ is not symmetric. I have already known that if matrix is symmetric then $A^{-1}$ is symmetric. But for this question I wrote the followings: If $A$ is not…
Fuat Ray
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Sum of diagonal elements of an odd-order latin square (matrix) symmetric about the leading diagonal with elements in $\{1,2,...,2n+1\}$

Let be a symmetric matrix of order $(2n+1)×(2n+1)$ which has on each row and each column every integer from $1$ to $2n+1$. What is the trace of the matrix? So, my guess is that the trace has every integer from $1$ to $2n+1$, so it is $(2n+1)(n+1)$…
Lucas McAllister
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Calculating the dimension of symmetric matrices.

Let $A=(a_{i,j})_{1\leq i,j\leq n}\in M_n(\mathbb{R})$ and $trace(A)=a_{1,1}+\cdots +a_{n,n}$. If $S_n(\mathbb{R}):=\left\{A\in M_n(\mathbb{R}):A^t=A\right\}$ and $\langle \cdot,\cdot \rangle:S_n(\mathbb{R})\times S_n(\mathbb{R})\to \mathbb{R}$…
eraldcoil
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Quadratic expression with any matrix has quadratic expression with symmetric matrix equivalent

Given $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ show that there exists a symmetric matrix $B \in \mathbb{R}^{n \times n}$ (which means $B^T = B$) for all A such that: $$x^TBx=x^TAx$$ This statement totally makes sense to me when I…
darisoy
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Minimizing the largest eigenvalue in a family of nonnegative matrices.

Let $\mathcal{F}$ be the family of symmetric, nonnegative matrices of order $n$ whose diagonal entries are zero. Let $\rho(A)$ be the largest eigenvalue of $A$. Suppose $\min\{\rho(A):A\in \mathcal{F}\}=\rho(B).$ Now consider a new family…
Sry
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