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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Is this matrix obviously positive definite?

Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1
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Properties of zero-diagonal symmetric matrices

I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices. The only thing I can come up with is that it cannot be positive definite (if it's not the zero matrix): The…
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Are there simple methods for calculating the determinant of symmetric matrices?

I've seen that there are lots of exercises about determinants of symmetric matrices in my algebra books. Some are easy and others are a bit more twisted, but the basic problem is almost always the same. I have been trying to come up with a method to…
user115173
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Are positive definite matrices robust to "small changes"?

Let $A$ be a positive-definite matrix and let $B$ be some other symmetric matrix. Consider the matrix $$ C=A+\varepsilon B. $$ for some $\varepsilon>0$. Is it true that for $\varepsilon$ small enough $C$ is also positive definite?
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Is $U=V$ in the SVD of a symmetric positive semidefinite matrix?

Consider the SVD of matrix $A$: $$A = U \Sigma V^\top$$ If $A$ is a symmetric, positive semidefinite real matrix, is there a guarantee that $U = V$? Second question (out of curiosity): what is the minimum necessary condition for $U = V$?
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Principal submatrices of a positive definite matrix

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k
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Why do positive definite matrices have to be symmetric?

Definitions of positive definiteness usually look like this: A symmetric matrix $M$ is positive definite if $x^T M x > 0$ for all vectors $x \neq 0$. Why must $M$ be symmetric? The definition seems to make sense for general square matrices.
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Analyze the symmetric property of positive definite matrices

In the definition, a positive definite matrix is usually referred to symmetric expressed in quadratic form. So I am confused about is it always symmetric? Why do they refer to the symmetric property in its definition? Please give me some examples…
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Interpretation of Symmetric Normalised Graph Adjacency Matrix?

I'm trying to follow a blog post about Graph Convolutional Neural Networks. To set up some notation, the above blog post denotes a graph $\mathcal{G}$, it's adjacency matrix $A$, and the degree matrix $D$. A section of that blog post then says: I…
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Eigenvalues of symmetric orthogonal matrix

Can we say that Eigenvalues of symmetric orthogonal matrix must be $+1$ and $-1$? Since eigenvalues of symmetric matrices are real and eigenvalues of orthogonal matrix have unit modulus. Combining both result eigenvalues of symmetric orthogonal…
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Decomposition of a positive semidefinite matrix

Let $Y \in \mathbb{R}^{n \times n}$ be a symmetric, positive semidefinite matrix such that $Y_{kk} = 1$ for all $k$. This matrix is supposed to be factorized as $Y = V^T V$, where $V \in \mathbb{R}^{n \times n}$. Does this…
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When is a symmetric matrix invertible?

My professor always writes on the board: $A$ is $m \times n$, assuming that the vectors of $A$ form a basis, then $A^TA$ is always invertible. one thing I know is that $A^TA$ is always symmetric, but I'm not sure about the conditions on a symmetric…
makansij
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"we note that the matrix Σ can be taken to be symmetric, without loss of generality"

I'm reading the book Pattern Recognition and Machine Learning by Christopher Bishop, and on page 80, with regard to the multivariate gaussian distribution: $$ \mathcal{N}(\mathbf{x} | \boldsymbol{\mu}, \boldsymbol{\Sigma}) = …
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Postitive definiteness of the Kronecker product of two positive definite matrices

Let $A$ and $B$ both be positive definite matrices. How do I show that their Kronecker product is also positive definite? I know we can use the fact that the eigenvalues of the Kronecker product is $\lambda_A+\lambda_B$ which are all positive. But…
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Can a symmetric matrix become non-symmetric by changing the basis?

We know that a hermitian matrix is a matrix which satisfies $A=A^*$, where $A^*$ is the conjugate transpose. A symmetric matrix (special case of hermitian - with real entries) is one for which $A=A^T$. Observation: this property is dependent on…