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Questions tagged [symmetric-matrices]

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

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32 views

skew symmetric matrix determinant always equal to 4 [on hold]

I have a question for homework. I tried to solve but I am not really successful about it. this is the question I have already create an example 4X4 matrix as an example but I can not make the proof. ...
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1answer
27 views

Largest eigenvalue of matrix product $A^T B A$

With $A \in \mathbb{S}^{d \times d}_+$ (symmetric positive semi definite) and $B \in \mathbb{S}^{d \times d}_{++}$ (symmetric positive definite), can we rewrite or upper bound $\lambda_{max}(A^T B A)$ ...
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1answer
26 views

How to prove the set of all symmetric matrices with eigenvalues in $(0,2)$ is connected?

Prove that space $X$ of all symmetric matrices in $GL_2(\mathbb R)$ with both the eigenvalues belonging to the interval $(0,2),$ with the topology inherited from $M_2(\mathbb R) $ is connected. ...
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2answers
52 views

How to show $\text{Tr}(M\log N)=\sum_{i,j}^n\lambda_i\log(\tilde{\lambda_j})(u_i^{\top}\tilde{u}_j)^2$?

The above question is the equation $(2.4)$ of the following paper: MATRIX EXPONENTIATED GRADIENT UPDATES. Let $M$ and $N$ be two $n \times n$ positive definite matrices where $M=U\Lambda U^{\top}$, $...
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1answer
47 views

Representation of negative Quantum entropy in terms of eigenvalues, i.e., $\text{Tr}(M\log M -M)=\sum_{i=1}^{n}(\lambda_i\log(\lambda_i)-\lambda_i)$?

Negative Quantum entropy or Negative Von Nuemann entropy is defined as $f(M)=\text{Tr}(M\log M -M)$. Where $M$ is a positive definite matrix in $\mathbb{S}_+^n$, $\log$ is natural matrix logarithm ...
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1answer
36 views

Determinant of a large symmetric block matrix

Consider a given matrix $Q \in \text{Mat}_N(\mathbb{R})$, which is invertible, and $n \geq 1$. I am looking for the determinant of the symmetric block matrix $I_n(Q)$ of total size $nN \times nN$: $$...
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1answer
37 views

How can I prove that $\mathbb{S}_{+}^n$ is a closed and convex set?

$\mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $n\times n$. I have to prove that this set is a closed convex cone. How can I do?
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1answer
21 views

quadratic programming /symmetric matrix

I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x. I can write Q as sum of ...
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2answers
17 views

implied eigenvalue equations for an arbitrary, symmetric and positive definite matrix

I have a matrix $M$, where $M$ = $\begin{bmatrix}a & b\\c & d\end{bmatrix}$. It is known that $M$ is symmetric and positive definite. Also, it is known that $x^TMy$ is a valid dot product in $...
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2answers
46 views

A special subset of $GL(n;\Bbb R)$

Let $M(n,\Bbb R)$ denote set of all $n\times n$ real matrices. For $A\in M(n,\Bbb R)$ we denote $A^t$ as the transpose of $A$. Denote $GL(n,\Bbb R)$ as set of all invertible real $n\times n$ matrices. ...
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2answers
318 views

Is the following matrix positive definite?

Is there an easy way to prove / disprove this? $$ (X)_{ij} = \begin{cases} \dfrac{k}{n}& \text{if}\ i = j \\ \dfrac{k(k-1)}{n(n-1)} & \text{otherwise} \\ \end{cases} $$ where $X \in \...
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1answer
24 views

Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...
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1answer
34 views

If A is a symmetric square matrix. I need to show that it is positive definite only if all eigenvalues are positive.

I understand that a positive definite matrix by the definition is a symmetric matrix where all eigenvalues are positive. I also know that if $ (x,y) = {x^T}{\cdotp}M{\cdotp}y$ then it is positive ...
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3answers
51 views

If I have a positive definite matrix X. How do i show that X$^2$ and X$^{-1}$ are also positive definite?

To my understanding a positive definite matrix is a real symmetric square matrix where all eigenvalues are positive. Therefore for a matrix A and vector v $Av = {\lambda}v$ where $\lambda$ is an ...
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1answer
30 views

How to show for any symmetric matrices the quadratic mean of eigenvalues less than square of Frobenius norm?

Let $A$ be a symmetric matrix which has $k$ non-zero eigenvalue. Show that the square of Frobenius norm is always bigger than the average of squared eigenvalues. That is: $$\|A\|_F^2 \geq \frac{1}{k} ...
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1answer
56 views

If $x$ and $y\in\Bbb{R}^{n}$ are eigenvectors for $\lambda\neq\mu$, respectively, show $x^{T}\cdotp y = 0$

For $x^{T}\cdotp y = 0$, I understand that I can either look at it through matrix multiplication $x^{T}y^{T} = 0$ as you can't do that multiplication. I'm very sure this isn't the right way of looking ...
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3answers
81 views

If $A=A^2$ is then $A^T A = A$?

I know that for a matrix $A$: If $A^TA = A$ then $A=A^2$ but is it if and only if? I mean: is this true that "If $A=A^2$ then $A^TA = A$"?
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Least squares and Gram matrix of B-spline derivatives

The Gram matrix of a B-spline basis is defined as $$ G_{ij} = \int_S B_i(x) B_j(x) dx $$ where the integral is taken over the full support $S$ of the B-splines. This matrix is positive definite and so ...
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2answers
41 views

How can we decompose the identity matrix given a set of orthonormal vectors?

Let $A$ be a positive semidefinite (P.S.D) matrix with distinct set of eigenvalues. since it is P.S.D its eigendecomposition is as follows for eigenpairs of $(\lambda_i,v_i)$ $$ A= \begin{bmatrix} ...
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0answers
13 views

Positive definite squares [duplicate]

Suppose that $A, B$ are real $n\times n$ symmetric positive definite matrices such that $A - B$ is positive semi-definite. Does it follow that $A^2 - B^2$ is positive semi-definite?
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Which of the following statements about determinants are correct?

Which of the following statements about determinants are correct? $\det(A^2)>0$, for all invertible matrices $A$ $\det(A+A^{-1})=\det(A)+\dfrac{1}{\det(A)}$, for all invertible matrices $...
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1answer
54 views

What is the solution for these two equalities $ x = \lambda(Ax) $ and $ x^TAx=1 $?

Let $A \in \mathbb{R}^{4 \times 4}$ be a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive. What is the solution for $x$ when $\lambda >0$ $$ x =...
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1answer
34 views

How to show that $\min_{x \in \mathbb{R}^4} x^Tx$ subject to $x^TAx \geq 1$ has a global minimizer?

Consider the following problem: $$\min_{x \in \mathbb{R}^4} x^Tx$$ over $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ where $A \in \mathbb{R}^{4 \times 4}$ is a symmetric matrix with two distinct ...
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2answers
45 views

Show that $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ is not empty?

Consider the set $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ where $A \in \mathbb{R}^{4 \times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are ...
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0answers
25 views

tensor power method

Just as we can use the matrix power method to find eigenvalues/eigenvectors of matrices in an iterative way, we can analogously find the eigenvalues/eigenvectors of tensors in a similar way. My ...
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0answers
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Explain: “SPD matrices can be thought of as an extension of positive numbers”

So I am reading a paper (https://www.researchgate.net/publication/263699451_From_Manifold_to_Manifold_Geometry-Aware_Dimensionality_Reduction_for_SPD_Matrices) during which the author states that "SPD ...
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0answers
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Interpretation of Symmetric Normalised of 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 ...
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6answers
302 views

Eigenvalues and Eigenvectors of Sum of Symmetric Matrix

Question: Let A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} Find all eigenvalues and eigenvectors of the martrix: $$\sum_{n=1}^{100} A^n = A^{100} +A^{99} +...+A^2+A$$ I know that ...
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0answers
18 views

Jacobi method- convergence.

I'm trying to find positive-definite matrix $A$ (3x3), such that Jacobi method is not convergent.
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2answers
24 views

Skew-symmetric non-diagonalizable matrix

Do you have an example of a real skew-symmetric matrix (seen as an operator over $\mathbb{C}^n$) having at least one (purely imaginary) eigenvalue with algebraic multiplicity strictly greater than the ...
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1answer
41 views

Questions on symmetric matrices and skew-symmetric matrices

Let $A$ be a $3\times 3\;$ symmetric matrix. Let $U$ be the set of all $3\times 3\;$ skew-symmetric matrices. Let $T : U\to U$ be defined as $T(B)=AB+BA.$ Prove that $T$ is bijective iff the sum of ...
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1answer
27 views

Symmetric matrices property

Reading "Mathematical Physics: Classical Mechanics" by A. Knauf, I found the following statement: The positive symmetric matrices with determinant 1 can be written as $$ \begin{vmatrix} A ...
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1answer
27 views

Why do these conditions ensure symmetric positive-definiteness?

Forgive me, if I have made a blunder or missed something obvious, I'm not a mathematician! I'm trying to understand 2 seemingly simple lines of maths - and understand how the conclusions are drawn ...
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3answers
140 views

How to solve $AX=XB$ for $X$ Matrix?

I have two symmetric $3\times 3$ matrices $A, B$. I am interested in solving the system $$AX= XB$$ Is there a way this is usually done? The matrices are not necessarily non singular.
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1answer
26 views

How do I find this eigenvector for a symmetric Matrix?

I have a symmetric matrix A, whose eigenvalues are $\lambda_1 = 6,~ \lambda_2 = 3,~ \lambda_3 = 2$ and eigenvectors are $\vec{v_1} = (1, 1, 1),~\vec{v_2} = (1,1,-1)$. How do I find the third ...
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1answer
75 views

Have I proven that every matrix is symmetric?

I've been given this as an assignment: "From definitions of different classes of matrices, prove the following claims: A) Positive => Symmetric…" There is also a hint: If $<Au,u> = <Bu,u&...
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1answer
39 views

Positive definite matrix with an interesting estimate [duplicate]

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a symmetric matrix. For all $i=1, \dots ,n$ we have $a_{ii} > \sum_{i \ne j} \vert{a_{ij}}\vert$. I now have to show that $A$ is positive definite. I ...
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1answer
20 views

Finding and creating Symmetric Matrix

Given the matrix, find symmetric closure of it. I am having a hard time understanding how to solve this. \begin{bmatrix}1&0&1&0&0\\1&1&0&1&1\\0&0&0&0&0\\...
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0answers
9 views

Convergence of an Iterative Method for Linear Systems

Consider the linear system $A \bf{x} = \bf{b}$, where $A$ is a symmetric matrix. Suppose that $M - N$ is a splitting of $A$ (i.e. $A = M - N$), where $M$ is symmetric positive definite and $N$ is ...
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4answers
269 views

Trace of symmetric matrix equals sum eigenvalues

I need to show that if $\mathbf{S}$ is symmetric, then it's trace sums to the sum of the eigenvalues. But I don't know how to show this. Can anybody give me a hint? P.S. Shame on my google skills, ...
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0answers
29 views

Real matrix decompose into symmetric matrices

Using Jordan standard form, we can prove Any complex matrix can be decomposed into the product of two symmetric matrices, and one of them is invertible. then how to prove that Any real square ...
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0answers
12 views

Sum of degenerate quadratic forms.

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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0answers
31 views

Bochner's theorem implies fourier transform of a pdf is a kernel?

I have just come across Bochner's theorem as applied to kernel functions. Does this reasoning follow? Consider $x_i$ and $x_j$ iid with pdf $f(x)$ and $x>0$. If one takes the Fourier transform of $...
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1answer
24 views

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$ I am studying now bilinear form .I wanted to prove above theorem. I ...
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2answers
47 views

Positive semidefiniteness of symmetric matrix with diagonal = 1 and non-diagonal elements less than 1

Does anyone know any useful results with respect to symmetric matrices with constant diagonals (specifically with respect to whether all eigenvalues are greater than $0$)? I am working on a set of ...
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0answers
28 views

Approximate symmetric matrix by minimizing condition number

We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number $\frac{\lambda_{\max}(X)}{\lambda_{\min}(X)}$. The optimization problem can be ...
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0answers
36 views

Are the eigenvectors of real Wigner matrices made of independent random variables with zero-mean?

I am trying to understand a portion of this paper [p. 3] and got stuck in the following statement, which sounded kind of trivial, but has been deceiving me for a while. Would you help me understand? ...
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1answer
40 views

Prove this matrix inequality

Let $W$ be $n$ by $n$ real symmetric positive definite matrix, let $V$ be an invertible $n$ by $n$ square matrix. Then the matrix $V^TWV$ is also symetric and positive definite. Now let $\|V\|$ denote ...
2
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1answer
49 views

Decomposing a positive semi-definite matrix with all -1,+1 elements

Claim.$\,$ A matrix $\,X \in \{-1,1\}^{n\times n}\,$ is positive semi-definite if and only if it is of the form $X= xx^{T}$, for some $x \in \{-1,1 \}^n$. How can I prove this? Proving the 'if' part ...
2
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0answers
56 views

Derivative of the matrix of eigenvalues of a real symmetric matrix

Given a real symmetric matrix $A$ with entries depending on $t$, the derivative $p$-th eigenvalue with respect to $t$ is given by $$ \lambda_p' = v_p^T A'v_p $$ where $A'$ denotes the derivative of ...