All Questions
Tagged with matrix-decomposition matrix-equations
237
questions
0
votes
0
answers
50
views
Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
- 49
1
vote
1
answer
30
views
Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
- 133
-2
votes
1
answer
81
views
What is the square root of a square matrix squared? [closed]
Admittedly, made the title a little funny, but this is a valid question.
I have come across the following equation
$$
I x^2=AA
$$
where $I$ is a unit matrix, $A$ is a square matrix of the same ...
- 101
0
votes
0
answers
70
views
Is there a closed form solution?
Question:
Let $A\in\mathrm{GL}(n,\mathbb{R})[[u]]$ with $A=A^T$, that is a symmetric real matrix, which can not be decomposed into a blockmatrix with $0$ blocks, such that
$$ A = \begin{pmatrix} A_1 &...
- 121
0
votes
0
answers
38
views
Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.
The problem is a conituation of this problem, but over finite fields.
In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
- 95
4
votes
1
answer
64
views
Binary matrix power for a specific entry.
Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$.
I am interested in the following decision procedure:
Given a ...
- 141
2
votes
1
answer
23
views
Equation for Counting Unique RREF "Cases" for any (m x n) Matrix
Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
- 23
0
votes
1
answer
43
views
Inverse $T$ matrix of a 3*3 matrix. [closed]
I have the matrix
$$ A=
\begin{bmatrix}
1 & 4 & 1\\
0 & 2 & 5\\
0 & 0 & 5
\end{bmatrix}.
$$
I have found the
$$ T=
\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 &...
- 9
1
vote
1
answer
44
views
Orthogonal procrustes with kernel constraint
Given a matrix $M \in \mathbb{R}^{n \times m}$ with $n > m$ and an arbitrary vector $v \in \mathbb{R}^n$, I am looking for an analytical solution for the orthogonal matrix $R \in \mathbb{R}^{n \...
- 421
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
0
answers
38
views
Solving Matrix Equation using SVD
I'm reading this paper by Bishop and Tipping.
They solve the equation
$$(SC^{-1} - I)W = 0$$
Where $W \in \mathbb{R}^{d \times q}$ and $S , C \in \mathbb{R}^{d \times d}$ and $C = WW^T + \sigma^2 I$ ...
- 43
0
votes
1
answer
31
views
Given general solution of a matrix, how to reconstruct its RREF form?
Here's the problem I am trying to solve:
I get that the second column would have an independent variable (as it would be a pivotal column) and the third and first columns would have nonpivotal ...
- 143
1
vote
0
answers
39
views
Eigenvalue-like problem for two matrices with coupled blocks
For given two real-symmetric matrices $A$ and $B$, both are of size $M \times M$, I'm trying to find two matrices $C_1$ (size $M\times N_1$) and $C_2$ (size $M\times N_2$) such that
$$
AC_1 = C_1\...
- 11
1
vote
0
answers
39
views
Howell form for symmetric matrices modulo N
I am looking at a problem of the type
$A^P\equiv A\pmod N$,
where A is a square symmetric integer matrix, and I want to find the minimum value of the power $P$. It seems that the Howell form of $A$ ...
- 111
2
votes
0
answers
24
views
How to solve the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ efficiently?
How would I go about solving the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ for $X$?
The simplest thing to do would be to, of course, consider $Y=\sum_iA_i^TC_iB_i^T$, vectorise and ...
