All Questions
Tagged with matrix-decomposition symmetric-matrices
109
questions
0
votes
1
answer
27
views
Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
- 149
0
votes
1
answer
23
views
For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
- 155
2
votes
0
answers
26
views
For a symmetric matrix $B$ and following four relevant matrices $P,Q,C,D$, what's the relation between $QP$ and $CDC^{\rm T}$?
Suppose $B\in\mathbb{R}^{n\times n}$ is a symmetric matrix with ${\rm rank}(B)=r$. Then $B$ is equivalent to $\tilde{B}$ in (1), where $I_{r}$ denotes the identity matrix of order $r$. That is, there ...
- 155
0
votes
0
answers
41
views
Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,497
0
votes
0
answers
24
views
Inverting a specific symmetric matrix preserves its zero entries
Suppose $S$ is an invertible symmetric matrix with the following property:
For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number ...
- 225
3
votes
1
answer
111
views
Solve $\| X A - B \|$ subject to $X C = C X$
Given $A, B \in \mathbb{R}^{n \times k}$ and S.P.D. $C \in \mathbb{R}^{n \times n}$, I would like to find an analytical solution for the matrix $X \in \mathbb{R}^{n \times n}$ that minimizes
\begin{...
- 421
0
votes
1
answer
105
views
Specific basis for the space of symmetric matrices
Consider the space of symmetric matrices $symm(M)$ over reals of dimension $n \times n$. It is clear that there is a straightforward basis for this space where for any $i \ge j$ $M_{ij}(m,n) = 1$ if $...
- 53
0
votes
0
answers
58
views
Zero trace matrix, product of symetric and antisymetric matrices
It is well-known that every square matrix can be written as the sum of a symetric matrix and an antisymetrix matrix.
The same does not hold for the product : for example, matrices $M\in\mathscr{M}_n(\...
- 243
1
vote
0
answers
49
views
Decomposition of a matrix into observability and controllability matrices
$\newcommand\iddots{\mathinner{
\kern1mu\raise1pt{.}
\kern2mu\raise4pt{.}
\kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.}
\kern1mu
}}$
I have a matrix $\boldsymbol{Q} \in \mathbb{R}^{M \times M}$ in ...
- 57
0
votes
1
answer
38
views
For a polar decompoistion $A=BJ$, the matrices $A$ and $B$ commute
Suppose $A$ is an invertible real $n\times n$ matrix, and consider its (unique) polar decomposition $A=BJ$ where $B$ is positive definite symmetric and $J$ is orthogonal. Is it true that $AB=BA$?
...
- 1,802
0
votes
0
answers
6
views
Which divergence or measure is more suitable for graph clustering application using symmetric NMF
Symmetric NMF is a well know tool used for graph clustering applications.
Given a similarity matrix $X \in \mathcal{R}^{n\times n}$, symmetric NMF seek to factorize it into o production of the form $...
- 562
1
vote
0
answers
48
views
Minimizing a log-determinant divergence based function
For a problem of the form $X\approx S$, where $X$ is an $n\times n$ symmetric but not always PD matrix, and $S$ its approximate, I am trying to minimize the log-determinant divergence between $X$ and $...
- 562
1
vote
0
answers
29
views
Name for 3 full diagonal matrix
I was wondering if there is a name or a terminology for a matrix with 3 full diagonals. It is similar with tridiagonal matrices, where the later is only centered at diagonal.
For such a matrix,
$$\...
- 337
2
votes
1
answer
156
views
When is a symmetric matrix $A ∈\Bbb R^{3×3}$ with three distinct eigenvalues equal to $A^n$?
How do you find a symmetric matrix $A ∈\Bbb R^{3×3}$, knowing that:
$A^n$ $= A$ for some $n > 1$
$A$ has three distinct eigenvalues
and we are given two (orthogonal) eigenvectors $[1, 2, 2]^T$ and ...
- 21
0
votes
0
answers
83
views
Distance between two positive semidefinite matrices
Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure ...
- 562