Questions tagged [symmetric-matrices]
A symmetric matrix is a square matrix that is equal to its transpose.
1,302
questions
1
vote
2answers
39 views
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 ...
0
votes
0answers
10 views
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-...
2
votes
1answer
40 views
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 ...
1
vote
1answer
57 views
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 ...
1
vote
0answers
29 views
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^\...
-1
votes
2answers
30 views
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 ...
1
vote
0answers
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 ...
2
votes
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:
...
3
votes
0answers
53 views
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\...
3
votes
1answer
67 views
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 $S$ ...
0
votes
0answers
45 views
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-...
1
vote
0answers
73 views
What is the $(i,j)'$th entry of the matrix?
Consider the symmetric matrix,
$$B = \begin{bmatrix}
4 & 60 & 360 & 840\\
60 & 840 & 4680 & 10080\\
360 & 4680 & 24480 & 50400\\
840 & 10080 & 50400 & ...
0
votes
2answers
45 views
Orthogonal Diagonalization of $~\begin{bmatrix} 9&-1&-2\\ -1&9&-2\\ -2&-2&6\end{bmatrix}~$ with repeated eigenvalue
Consider a matrix
$$
A=\begin{bmatrix} 9&-1&-2\\ -1&9&-2\\ -2&-2&6\end{bmatrix}.
$$
We now find the orthogonal matrix $P$ such that $P^TAP$ is diagonal.
My attempt:
Eigenvalues ...
0
votes
0answers
18 views
How to find the iteration no. with precision of a preconditioned CG method?
So to solve the equation $Ax = b$, the original discrete matrix is
$$A = \operatorname{tridiag}(-a_i, a_i + a_{i+1}, -a_{i+1}) \in \mathbb{R}^{n \times n},$$
$i$ from $0$ to $n, 0 < C_1 \leq a_i \...
3
votes
0answers
22 views
Is the expectation of the inverse of a random matrix that has diagonal expectation also diagonal?
Suppose we have a symmetric positive definite random matrix $\mathbf{A}$ that has a diagonal matrix expectation,
$$
\mathbf{E}(\mathbf{A}) = \text{diag}(\mathbf{x}),
$$
[like e.g. the Wishart ...
-1
votes
1answer
63 views
Explanation on a theorem of linear algebra
"Let $A$ be a $n\times n$ matrix with a orthonormal basis of $n$ eigenvectors, show that $A$ is symmetric."
No idea how to prove this.
0
votes
1answer
29 views
Coerzive bilinear forms in Sobolev Space $H^1(\Omega)$.
Consider $\Omega=[0,1]^3$ and the bilinear form
$B(u,v)=\int_{\Omega} (\nabla u)^T A \nabla v +uv \,dx, \; \; u,v \in H^1(\Omega).$
The matrix $A \in \mathbb{R}^{n \times n}$ is symmetric and positive ...
0
votes
1answer
36 views
Square matrices such that $A = B + C$ with each matrix diagonalisable. Must $B$ and $C$ commute?
Suppose $A, B, C$ are square, real, symmetric, positive semi-definite matrices such that
$$
A = B + C
$$
and $A$ is positive definite.
Must $B$ and $C$ be simultaneously diagonalisable?
I'm thinking ...
2
votes
0answers
18 views
Interpretation of Symmetric Normalised Weighted Adjacency Matrix in GCN
Yann Dubois explained very well about the "Interpretation of Symmetric Normalised Graph Adjacency Matrix?".
He defined that adjacency matrix  can be weighted. He also defined that D̂ is ...
1
vote
2answers
36 views
How to derive a constraint for a positive semi-definite matrix?
Given a positive semi-definite (PSD) matrix $M$ =:
$$\begin{bmatrix}
1 & a & c \\
a & 1 & b \\
c & b & 1
\end{bmatrix}$$, how to come up with the constraint:
$$ab - ...
0
votes
0answers
24 views
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 ...
2
votes
3answers
69 views
$A$ 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,\...
0
votes
0answers
23 views
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 $\...
0
votes
0answers
55 views
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 ...
1
vote
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?
0
votes
1answer
21 views
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 $i \neq j$,...
3
votes
0answers
40 views
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 ...
1
vote
0answers
20 views
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:
$$\sum_{j=...
1
vote
1answer
43 views
Infimum over $u$ of $(Mw+Pu)^TA(Mw+Pu)$ is $u=-(P^TAP)^{-1}P^TAMw$
Let $A$ be symmetric positive definite. Let $M$ and $P$ be full rank matrices such that every vector $v$ can be uniquely decomposed as $v=Mw+Pu$ and the matrix $[M,P]$ is invertible.
$P(P^TAP)^{-1}P^...
1
vote
2answers
27 views
Why is $Av_j=\lambda_jv_j$?
I don't quite understand the solution give to this exercise, so I would like some clarification on that:
Let $v_1,...,v_p \in \mathbb{R}^p$ be orthonormal vectors, and for some $$-1< \lambda_p<\...
2
votes
1answer
37 views
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 ...
1
vote
2answers
45 views
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 ...
0
votes
0answers
26 views
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 ...
5
votes
1answer
29 views
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 ...
-1
votes
1answer
43 views
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 ...
0
votes
0answers
17 views
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. ...
0
votes
0answers
21 views
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 &...
0
votes
0answers
34 views
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$ $\...
0
votes
0answers
31 views
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 ...
4
votes
1answer
76 views
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 ...
0
votes
0answers
15 views
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$
$\...
-1
votes
1answer
35 views
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 ...
1
vote
0answers
45 views
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$...
2
votes
1answer
51 views
$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 \...
0
votes
0answers
34 views
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 ...
4
votes
1answer
39 views
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 ...
2
votes
3answers
85 views
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 $L$ ...
1
vote
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 $B= B^{1/2}B^{1/2}$, ...
1
vote
0answers
46 views
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. ...
4
votes
2answers
95 views
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 $...
