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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.

1974 questions
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Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where…
Phonon
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Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, but it has been a long time, and I can't find it in…
gregmacfarlane
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Why is $A^TA$ invertible if $A$ has independent columns?

How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar question, phrased the other way around, so I tried to use the theorem $$ rank(A^TA) \le min(rank(A^T),rank(A)) $$ Given $rank(A) = rank(A^T) = n$ and…
Chewers Jingoist
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What is the largest eigenvalue of the following matrix?

Find the largest eigenvalue of the following matrix $$\begin{bmatrix} 1 & 4 & 16\\ 4 & 16 & 1\\ 16 & 1 & 4 \end{bmatrix}$$ This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and,…
Cloud JR K
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Can a real symmetric matrix have complex eigenvectors?

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and real eigenvalues, but could it ever have complex…
Phil H
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How to generate random symmetric positive definite matrices using MATLAB?

Could anybody tell me how to generate random symmetric positive definite matrices using MATLAB?
Srijan
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Prove that the eigenvalues of a real symmetric matrix are real

I am having a difficult time with the following question. Any help will be much appreciated. Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the eigenvalues of a real symmetric matrix are real…
Susan
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The inverse of a positive definite matrix is also positive definite

Let $K$ be nonsingular symmetric matrix, prove that if $K$ is positive definite so is $K^{-1}$ . My attempt: I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next.
diimension
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Do positive semidefinite matrices have to be symmetric?

Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as…
Molly
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Which polynomials are characteristic polynomials of a symmetric matrix?

Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic polynomial is $f$. My question is: when is it possible…
Paul Siegel
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Is the sum of positive definite matrices still positive definite?

I have two symmetric positive definite (SPD) matrices. I would like to prove that the sum of these two matrices is still SPD. Symmetry is obvious, but what about PD-ness? Any clues, please?
Richard Dong
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Norm of a symmetric matrix equals spectral radius

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an operator in $\mathbb{R}^n$ given by $x \mapsto Ax$. Prove…
Erik Vesterlund
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Relationship between eigendecomposition and singular value decomposition

Let $A \in \mathbb{R}^{n\times n}$ be a real symmetric matrix. Please help me clear up some confusion about the relationship between the singular value decomposition of $A$ and the eigen-decomposition of $A$. Let $A = U\Sigma V^T$ be the SVD of…
user8478
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Dimensions of symmetric and skew-symmetric matrices

Let $\textbf A$ denote the space of symmetric $(n\times n)$ matrices over the field $\mathbb K$, and $\textbf B$ the space of skew-symmetric $(n\times n)$ matrices over the field $\mathbb K$. Then $\dim (\textbf A)=n(n+1)/2$ and $\dim (\textbf…
Christian Ivicevic
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Is there a fast way to prove a symmetric tridiagonal matrix is positive definite?

I' m trying to prove that $$A=\begin{pmatrix} 4 & 2 & 0 & 0 & 0 \\ 2 & 5 & 2 & 0 & 0 \\ 0 & 2 & 5 & 2 & 0 \\ 0 & 0 & 2 & 5 & 2 \\ 0 & 0 & 0 & 2 & 5 \\ \end{pmatrix}$$ admits a Cholesky decomposition. $A$ is symmetric, so it admits a…
Yagger
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