# Questions tagged [symmetric-matrices]

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

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### Zero set of system of two real quadratic forms

Background: Consider the equation $x^T A_1 x = 0$ where $x \in \mathbb{R}^\mu$ and $A_1 \in \mathbb{R}^{\mu \times \mu}$ is a symmetric matrix. Suppose we also demand the normalization $x^T x = 1$. ...
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### Effect of Scaling a Row and Column on Multiplicities of Nonzero Eigenvalues of a Symmetric Matrix

Question: Let $A$ be a real symmetric $n\times n$ matrix with eigenvalues $\lambda_1, \lambda_2,\cdots, \lambda_n$ and corresponding algebraic multiplicities $a_1, a_2, \cdots, a_n$. Consider a ...
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### Understanding an Inequality In a Paper

The paper is the following: https://arxiv.org/pdf/1608.06412.pdf Right above (5) on page 5, the first inequality of the line where they use lemma 8, I'm not quite sure I see how they are using lemma 8....
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### What's the best way to make a symmetric matrix positive definite?

Assume that you have a matrix $X \in \mathbb R^{m \times m}$ and it's symmetric, but it's not positive definite. What's the best way to turn the matrix $X$ into a positive definite matrix? I have a ...
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### On the $\log \det$ of identity matrix plus a symmetric positive definite matrix

I am trying to learn some matrix differentiation, and came across example of calculating the derivative of $$f(X)=\log\det(X)$$ where the $X$ is a symmetric positive definite matrix. I came to the ...
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### Is there a case in which PSD matrices $A$ and $B$ satisfy the condition $A \approx B$ but $A^2 \not \approx B^2$?

For two PSD matrices $A, B$ and a positive number $\alpha$, let's define $A \approx_{\alpha} B$ as $A \preceq \alpha B$ and $B \preceq \alpha A$, where $A \preceq B$ means $B - A$ is PSD. And if there ...
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### Existence of a real symmetric matrix similar to a Jordan block corresponding to 0

Let $n$ be a positive integer. Let $J_n(0)$ denote the $n$-by-$n$ Jordan block corresponding to 0. For each $n$, does there exist a real symmetric matrix $B$ such that $B$ is similar to $J_n(0)$? ...
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### If $A$ is positive real symmetric matrix and $B$ is real symmetric, does there exist some $V$ such that $V^{T}AV$=$I$ and $V^TBV$ is diagonal matrix?

Let's say we have two matrices $A$ (a positive real symmetric matrix) and $B$ (a real symmetric matrix). And let us suppose that in general $A$ and $B$ don't commute with each other. Then, Q. Is it ...
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### $A,B$ positive semidefinite matrices with $A \geq B$ implies $A^2 \geq B^2$? [duplicate]

Is it true that if I am considering positive semidefinite matrices $A, B$ with $A \geq B$ then $A^2 \geq B^2$? Could you help me prove this or think of a counterexample (eventually assuming the ...
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### If $A$ is symetric and positive definite, when will $A + B$ be invertible?

$A$ is a symmetric and positive definite matrix with size of $n \times n$, and $B$ is a matrix of the same size. Under which circumstance will $A+B$ be invertible?
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### Iterative algorithm for computing $\Sigma^{1/2} x$

Say I have a PSD matrix $\Sigma$ and a vector $x$, is there an iterative algorithm (faster than computing $\Sigma^{1/2}$ using Cholesky decomposition) for computing $\Sigma^{1/2} x$? (In this ...
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### Projecting a symetric matrix on a subspace of known kernel

Given a symmetric matrix ${\bf S} \in S_n(\mathbb{R})$ and linearly independent vectors ${\bf u}, {\bf v}, {\bf w} \in \mathbb{R}^n$, how can one numerically compute the projection of $\bf S$ onto the ...
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### Why does Wolfram say this symmetric matrix has complex eigenvalues?

According to Wolfram, the following matrix has complex eigenvalues. Symmetric matrices have real eigenvalues, so I’m not sure what I’m failing to understand. The matrix is the shape operator of the ...
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### Prove that condition number of a matrix increases in its dimension

This is my first time asking a question here. Thus, I apologise in advance if it is not articulated correctly or something else turns out to be wrong with it. Before asking the question itself, ...
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### How to calculate the differentiation of this expressions which contains matrix variable?

it is assumed that $\mathbf{X}\in\mathbb{R}^{n\times n}$ is symmetric, and $\mathbf{Y}=\mathbf{M}^\textsf{T}{\mathbf{X}}^{-1}\mathbf{M}$, where $\mathbf{M}\in\mathbb{R}^{n\times m}$ has no special ...
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### Eigenvalues of a squared symmetric matrix

In Page 185 here it says ... $M^2 y=\sigma^2y$. Since $M$ is symmetric, it follows that $y$ is an eigenvector of $M$ with eigenvalue $\pm \sigma$. It seems to contradict the example here. What am I ...
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### Definition of the (anti-)symmetric part of a tensor

Consider the following tensor: $$A_{ij} = B_{k,i}C_{k,j}$$ which can be decomposed into its symmetric and antisymmetric parts: $A_{ij} = A_{ij}^s + A_{ij}^a$. I'm all confused as to how to express ...
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### What are the minimal conditions for this matrix to be invertible?

Let $A>0$ be a $n \times n$ symmetric positive definite (SPD) and $B$ be an $n \times m$ matrix, then form the matrix $$C = \left[\begin{array}{cc} A & B \\ B^T & 0\end{array}\right],$$ ...
I am struggling to understand a proof of theorem 3.5.6 in Roman Vershynin's High-Dimensional Probability Theorem: \text{INT}(A) = \max_{x_i = \pm 1 \text{ for } i = 1,\ldots,n} \sum_{i,j=1}^{n} A_{...
### How do I prove that for any $2\times2$ matrix $A$, $A^2$ can be written in the linear form $aA + bI$ where $a$ and $b$ are scalars? [closed]
How do I prove that for any $2\times2$ matrix $A$, $A^2$ can be written in the linear form $aA + bI$ where $a$ and $b$ are scalars? I've tried letting the elements of A be $w$, $x$, $y$, and $z$ then ...