Questions tagged [symmetric-matrices]

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

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Is a symmetric real matrix similar to a diagonal matrix through an orthogonal matrix?

Definition Two matrices $A$ and $B$ are said similar if there exist an inverible matrix $P$ such that $$ B=PAP^{-1} $$ Definition A square matrix $A$ is said orthogonal if it is invertible and its ...
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Ordered eigenvalues of circulant matrix when using DFT

Let $\mathbf{C} \in \mathbb{R}^{n \times n}$ be a circulant matrix: \begin{equation} \mathbf{C} = \begin{pmatrix} {c_0} & {c_1} & {\dots} & {c_{n-2}} & {c_{n-1}} \\ {c_{n-...
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eigenvectors of a real symmetric matrix are always orthogonal

As we know by the famous theorem "eigenvectors corresponding to distinct eigenvalues are orthogonal for a real symmetric matrix" can this result be also true for the same eigenvalues My ...
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Given the distance measure for two points and the point $p = (0,2)$, which of the following points have the same distance as $p$ from the origin?

Given the distance measure dist: $$\mathrm{dist}(x,y) = \sqrt{(x_1-y_1,x_2-y_2)\begin{pmatrix} 3 & 0\\ 0 & 4 \end{pmatrix} (x_1-y_1,x_2-y_2)^{T} } $$ for two dimensional points and the ...
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ODE for the Cholesky decomposition of a SPD matrix

Let $P$ be a spd matrix and consider the ODE $$ \frac{dP}{dt} = FP + PF^\top + C.$$ The ODE makes sure that $P(t)$ is spd for any $t$. I would prefer to obtain the Cholesky decomposition of $P = L L^\...
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How to calculate the dimension of a space that is defined by symmetric matrices [duplicate]

Let $M = \{ A \in \mathbb{R}^{n \times n} | A^{T} = A \}$ be the space of symmetrical matrices. How do I calculate the dimension of this space? As far as I understand the dimension of a space is equal ...
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39 views

Inequality quadratic form

Let $A$ be symmetric positive definite, let $B$ and $C$ be rectangular matrices such that there is a decomposition $v=Bu+Cw$ for every vector $v$. Let $p$ be a projection with $Range(p)=Range(C)$ and ...
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1answer
115 views

Eigenvector of A by $A^2$

Let $A$ be a real symmetric matrix. Assume $A^{2}v=\lambda Av$ , can we somehow deduce $v$ is an eigenvector of $A$ with $\lambda$ as its corresponding eigenvalue? i.e $Av=\lambda v$. Thanks Edit: ...
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Perron eigenvector of symmetric matrix

We have a "wide" matrix $A\in\mathbb{R}_+^{k \times n}$ where $n$ is usually large (e.g. 10000) and $k$ is small (e.g. 5 to 10). All of its entries are positive. Now, we introduce $B=AA^T\in\...
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If $S$ is symmetric positive definite and $SA$ symmetric, is then $A$ symmetric?

We are given real matrices $S$ and $A$. We know that $S$ is symmetric positive definite and that $SA$ is symmetric. Is A necessarily symmetric then? I've figured out that if $A$ is symmetric, then $S$ ...
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Invariants of a symmetric bilinear form

The following is a theorem from the Linear Algebra Textbook by Friedberg, Insel, and Spence (5th Edition). Theorem 6.38 (Sylvester's Law of Inertia). Let $H$ be a symmetric bilinear form on a finite-...
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What is the $(i,j)'$th entry of the matrix?

Consider the symmetric matrix, $$B = \begin{bmatrix} 4 & 60 & 360 & 840\\ 60 & 840 & 4680 & 10080\\ 360 & 4680 & 24480 & 50400\\ 840 & 10080 & 50400 & ...
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Orthogonal Diagonalization of $~\begin{bmatrix} 9&-1&-2\\ -1&9&-2\\ -2&-2&6\end{bmatrix}~$ with repeated eigenvalue

Consider a matrix $$ A=\begin{bmatrix} 9&-1&-2\\ -1&9&-2\\ -2&-2&6\end{bmatrix}. $$ We now find the orthogonal matrix $P$ such that $P^TAP$ is diagonal. My attempt: Eigenvalues ...
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How to find the iteration no. with precision of a preconditioned CG method?

So to solve the equation $Ax = b$, the original discrete matrix is $$A = \operatorname{tridiag}(-a_i, a_i + a_{i+1}, -a_{i+1}) \in \mathbb{R}^{n \times n},$$ $i$ from $0$ to $n, 0 < C_1 \leq a_i \...
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Is the expectation of the inverse of a random matrix that has diagonal expectation also diagonal?

Suppose we have a symmetric positive definite random matrix $\mathbf{A}$ that has a diagonal matrix expectation, $$ \mathbf{E}(\mathbf{A}) = \text{diag}(\mathbf{x}), $$ [like e.g. the Wishart ...
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Explanation on a theorem of linear algebra

"Let $A$ be a $n\times n$ matrix with a orthonormal basis of $n$ eigenvectors, show that $A$ is symmetric." No idea how to prove this.
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Coerzive bilinear forms in Sobolev Space $H^1(\Omega)$.

Consider $\Omega=[0,1]^3$ and the bilinear form $B(u,v)=\int_{\Omega} (\nabla u)^T A \nabla v +uv \,dx, \; \; u,v \in H^1(\Omega).$ The matrix $A \in \mathbb{R}^{n \times n}$ is symmetric and positive ...
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Square matrices such that $A = B + C$ with each matrix diagonalisable. Must $B$ and $C$ commute?

Suppose $A, B, C$ are square, real, symmetric, positive semi-definite matrices such that $$ A = B + C $$ and $A$ is positive definite. Must $B$ and $C$ be simultaneously diagonalisable? I'm thinking ...
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Interpretation of Symmetric Normalised Weighted Adjacency Matrix in GCN

Yann Dubois explained very well about the "Interpretation of Symmetric Normalised Graph Adjacency Matrix?". He defined that adjacency matrix  can be weighted. He also defined that D̂ is ...
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How to derive a constraint for a positive semi-definite matrix?

Given a positive semi-definite (PSD) matrix $M$ =: $$\begin{bmatrix} 1 & a & c \\ a & 1 & b \\ c & b & 1 \end{bmatrix}$$, how to come up with the constraint: $$ab - ...
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Show proofs for inverse of these singular matrix.

I tried really hard, but I have no idea how to approach this question. A and B matrix are not invertible, so inverse does not exist. So, how do I go about proving them ? Simply saying they do not have ...
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$A$ is positive semidefinite $\iff \text{det} (B_K) \geq 0$

Let $A \in \mathbb R^{n \times n}$ a symmetric matrix. Show that $A$ is positive semidefinite $\iff$ all its symmetric minors are $\geq 0$, that means $\det(B_K) \geq 0$ for all $K \subseteq \{1,\...
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vecp vs vech operator on symmetric matrices

Let $\mathbf{X}$ $(p \times p)$ be a symmetric matrix. The vecp operator stacks the elements of $\mathbf{X}$ above and including the diagonal columnwise. The vech operator stacks the elements of $\...
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Why is the derivative of $\frac{\partial{Tr(AXB)}}{\partial{X}}$ for symmetric matrix X equal to the asymmetric result?

I'm a bit confused of the fact that an online matrix calculus calculator (https://www.matrixcalculus.org/matrixCalculus) gives the same result for the derivative w.r.t. a symmetric matrix $X$ and a ...
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1answer
60 views

Inequality between symmetric matrices

Let $N$ be symmetric positive definite with $N-S$ positive semidefinite and let $Y := N (2N -S)^{-1} N$. Then $\frac{1}{2} v^{\top} N v \leq v^{\top} Y v$. How does this follow?
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Bounds on extremal eigenvalues of gram matrix with diagonal entries in $[a, b]$ and off-diagonal entries in $[c, d]$

Consider a square $n \times n$ matrix $H = A^T A$ where $A$ is an $m \times n$ with $m \ge n$. Knowing that $a \le H_{ii} \le b$ for all $i = 1, \ldots, n$ and that $c \le H_{ij} \le d$ for $i \neq j$,...
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On the Symmetrized Product $AB+BA$

I have encountered the matrix $P=P^T$, \begin{align*} P:=AB+BA, \end{align*} where $A,B\in\mathbb{R}^{n\times n}$ and (in my case) $A=A^T$, $B=B^T$ and $1 \succeq A,B\succeq 0$. The matrix $P$ is ...
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Eigenvalues and other links between triangular and corresponding symmetric matrix

In Maths and Physics we often stumble across problems that can be described either by a triangular matrix or by a symmetric matrix. In particular it is usual to meet sums like the following: $$\sum_{j=...
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1answer
43 views

Infimum over $u$ of $(Mw+Pu)^TA(Mw+Pu)$ is $u=-(P^TAP)^{-1}P^TAMw$

Let $A$ be symmetric positive definite. Let $M$ and $P$ be full rank matrices such that every vector $v$ can be uniquely decomposed as $v=Mw+Pu$ and the matrix $[M,P]$ is invertible. $P(P^TAP)^{-1}P^...
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2answers
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Why is $Av_j=\lambda_jv_j$?

I don't quite understand the solution give to this exercise, so I would like some clarification on that: Let $v_1,...,v_p \in \mathbb{R}^p$ be orthonormal vectors, and for some $$-1< \lambda_p<\...
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1answer
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About real symmetric matrix multiplied by diagonal matrix [closed]

Recently, I found an important matrix in analog circuit domain and it need to be proved diagonalized. Then I try to resolve it into a small problem that is: if there are a $n\times n$ real symmetric ...
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2answers
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strictly diagonally dominant by rows matrix and eigenvalues

I have a question about this exercises: We suppose that we have a matrix A with real eigenvalues $λ_1 > · · · > λ_n> 0$ and strictly diagonally dominant by rows such that: $$ \gamma \lvert ...
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Why is the covariance of an Ornstein-Uhlenbeck process symmetric and positive semi-definite?

I have been working with the d-dimensional Ornstein-Uhlenbeck process $dX_t^{\epsilon}=-QX_t^{\epsilon}dt + \epsilon dB_t$, $X_0^{\epsilon}=x \in \mathbb{R}$. I know that, by Ito's formula, you can ...
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What is a non binary adjacency Matrix?

I am going through the implementation of a graph convolutional neural network. I came across a non-binary adjacency matrix in the case of a directed graph. The particular issue is discussed here in ...
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Product of the inverse of a positive definite matrix and a symmetric matrix [closed]

Let P be a symmetric positive definite matrix and A is a symmetric matrix (with A and P real-valued matrices). Show that $P^{-1} A$ is diagonalizable and its eigenvalues are all real. I tried to show ...
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Looking for a graph with the biggest absolute eigenvalue of adjacency matrix negative.

In my class, our teacher define the spectral radius to be the largest eigenvalue of the adjacency matrix, while I find in wiki that the spectral radius is defined as the largest absolute eigenvalues. ...
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When one eigenvalue equals the sum of the others?

I am reading a paper that based on the characteristics of the symmetric matrix $G$ derives a property of the eigenvalues of the gramian matrix $G^TG$. Since \begin{equation} G = \begin{pmatrix} x_1 &...
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Orthogonal matrix $O$ maximising $Tr(OM)$.

One can prove that the set $\{ Tr(OM) /$ $O$ $\in$ $ O_n ( \mathbb R ) $ $\}$ has a maximum for a certain orthogonal matrix $O$, furthermore, the application $f$ : $M_n (\mathbb R)$ $\mapsto$ $\...
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It seems that minimizing the largest eigenvalue of this symmetric positive definite matrix maximizes its trace. How?

Starting from the matrix \begin{equation} H = \begin{pmatrix} x_1 & y_1 & z_1 & 1 \\ \vdots & \vdots & \vdots & \vdots \\ x_n & y_n & z_n & 1 ...
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1answer
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How can we state Sylvester's law of inertia without referring to a particular basis?

In elementary linear algebra, we talk about matrices, i.e. rectangular arrays of numbers. In advanced linear algebra, we prefer whenever possible to talk about abstract tensors, such as linear ...
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2x2 symetric markov matrix A and Y a vector with entries between 0 and 1, is X a vector with entries between 0 and 1 in AX = Y?

I have a matrix $ A = \left[ {\begin{array}{cc} w & 1-w \\ 1-w & w \\ \end{array} } \right] $ with $0<w<1/2$ I have $AX = Y$ with $Y = (y_1, y_2)$ with $0\leq y_i \leq 1$ $\...
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Commutativity of Positive Definite Matrices

How do you show that two matrices, $A$ and $B$ that are positive definite commute? I know this property is true since the set of positive matrices of size n, $Pos(n,\mathbb{R})$ is a subgroup of ...
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Diagonalization of a real symmetric matrix

I saw in the spectral theorem that any real symmetric matrix $A$ is diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix $A$, $Q^{T} AQ$ is diagonal for some orthogonal matrix $Q$...
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1answer
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$A$ is Positive Definite iff $B=\frac{1}{2}\left(A+A^{\mathrm{T}}\right)$ is positive definite

While studying Symmetric matrices for my nonlinear optimization class, I encountered the following problem: Problem: Suppose that $A$ is an arbitrary real square matrix. Show that $\mathbf{x}^{T} A \...
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Why does Householder tridiagonalization cost $4n^3/3$?

In Golub, Van Loan - Matrix Computations page 459, it says This algorithm requires 4n^3/3 flops when symmetry is exploited in calculating the rank- 2 update. But I don't quite understand that. How ...
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1answer
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Eigenvalues of the product of symmetric and positive definite matrices

Let $A$ , $B$ be two real symmetric matrices and $A$ is positive definite. Then show that $AB$ has real eigenvalues. Symmetric matrices have real eigenvalues and product of two symmetric matrices need ...
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3answers
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show that symmetric and anti-symmetric matrices are eigenvectors for linear map

Consider the linear map $$L:Mat_{nxn}(\mathbb{R})\rightarrow Mat_{nxn}(\mathbb{R}),\\ M\mapsto 4M-7M^T$$ a) Prove that any symmetric or anti-symmetric matrix is an eigenvector for $L$. b) Is $L$ ...
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1answer
63 views

Spectral radius and positive definite matrix

Let $A$ be a $m \times m$ matrix. Show that there exists a $m \times m$ positive definite matrix $B$ such that $B - A^H B A \succ 0$ if the spectral radius $\rho(A) < 1$ Let $B= B^{1/2}B^{1/2}$, ...
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Measurability of a solution to $Ax=b$ for measurable symmetric matrix

Let $(\Omega, \mathcal F) $ be a measurable space; $A:\Omega \to \mathbb R^{n\times n} $ be a measurable matrix-valued function and $b:\Omega\to \mathbb R^n$ be a measurable vector-valued function. ...
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2answers
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If $A<B$, does it follow that $A^2\leq B^2$?

Suppose $A$ and $B$ are positive semidefinite matrices satisfying $A\leq B$ (meaning that $x^TAx \leq x^TBx$ for any vector $x$). Does it follow that $A^2\leq B^2$? It certainly follows if $A$ and $...

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