All Questions
Tagged with matrix-decomposition matrix-calculus
185
questions
2
votes
0
answers
34
views
need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
- 41
1
vote
0
answers
30
views
Optimization of eigenvalue of matrix with discrete variables
Suppose we have a matrix $H$ like this, where $H_{ii}$ is a discrete variable (for example out of $[1,2,3]$) and $H_{ij}$ is a value that depends on the two adjacent values ($H_{ii}$ and $H_{jj}$). $H$...
- 31
1
vote
0
answers
78
views
Symplectic approximation to a given matrix
I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
- 61
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
$$ T=
\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 &...
- 9
1
vote
1
answer
73
views
Derivative of the Cholesky factor
I have a symmetric positive definite matrix $L\in\mathbb{R}^{n\times n}$ and its Cholesky factor $G\in\mathbb{R}^n$ such that $L=GG^T$. Called $\mathrm{vec}:\mathbb{R}^{n\times n}\to\mathbb{R}^{n^2}$ ...
- 846
3
votes
0
answers
48
views
When do symmetrical matrices commute?
For context I was taking an optimization course and at one point we used the restriction by line to prove the concavity of $\log det(X)$. Let $g(t) = \log det(X + tV)$ with $V \in S_{n}$ and $t \in \...
- 31
0
votes
0
answers
23
views
Unitary Freedom, PSD matrices
Suppose I have a matrix $ Y \in \mathbb{R}^{N \times T}$ where each column $y_t \in \mathbb{R}^{N}$ is a measurement of $N$ different quantities. Assuming the matrix $Y$ is zero-mean on each row, we ...
- 888
3
votes
0
answers
98
views
Weighted Nearest Kronecker Product
For a given $m n$-length vector $a$, the problem of finding an $m$-length vector $x$ and an $n$-length vector $y$ that minimize
$$
\lVert a - x \otimes y \rVert^2
$$
is known as the Nearest Kronecker ...
1
vote
0
answers
77
views
Can I use QR decomposition as a last step in a Constrained Orthonormal Matrix Optimization problem?
I have the optimization problem: minimise $f(V)$, where $V$ is $N\times N$, subject to
$V$ is orthonormal
All entries of the first column of $V$ are $1/\sqrt{N}$
$V \cdot D \cdot V^T \cdot \mathbf{1}...
- 379
0
votes
1
answer
161
views
Algorithm for Deciding if a Quadratic Form is Positive Definite
I am hoping to find a general algorithm to decide if a quadratic form is positive definite. My approach has been as follows:
Let $q$ be a quadratic form, and let $q(h) = h^{T}Ah$. Since $A$ is ...
- 402
1
vote
0
answers
59
views
The matrix inverse of $\begin{pmatrix} a_{11} 10^{10}& a_{12} 10^{30} \newline a_{21} 10^{20}& a_{22} 10^{40} \end{pmatrix} $ and an algebra?
The question started with how to compute the matrix inverse with a scale separated orders.
A related question could be found here:
Compute matrix inverse at scale separated orders
This is, in $2\times ...
0
votes
1
answer
118
views
Square root of PSD matrix and unitary freedom
Consider a given PSD matrix $S$, which can be written as:
$$S= A A^\top$$
where $A$ is its unique PSD square root. I am computing $A$ through the EVD of $S= U DU^\top$, i.e., $A= U D^{1/2}U^\top$.
I ...
- 888
0
votes
0
answers
21
views
How to solve the system of equation $y=\operatorname{Diag}(Bz) x $ with respect $B$ and z
Assume I have the following system of equations:
$$y = \operatorname{Diag}(h) x$$
where $x,y, h \in {\bf R}^N$ and suppose that I know them.
I also know that $h$ can be factorized as: $$h=Bz$$
where $...
- 888
0
votes
1
answer
76
views
Maximization on trace of quadratic and linear terms under orthonormal constraints
I have the following optimization problem
$$ \max_{R: RR^{T}=I} \mbox{Tr} \left( M \left( R A R^{T} - K R^{T} \right) \right) $$
where:
$A$ is a rank-one square matrix (assume the first row that are ...
- 31
0
votes
0
answers
37
views
Necessary condition for $X^{\prime}AX$ to be inversible
Let $A$ be the positive definite matrix. I used the inverse of $X^{\prime}AX$ somewhere in my calculations, for some real-valued matrix $X$, and I set the assumption that $X^{\prime}AX$ is a ...
- 473