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Results tagged with matrices
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user 50191
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
4
votes
1
answer
2k
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Derivative of trace of pseudo inverse
Given three matrices $A$ (broad), $B$ and $C$, I'd like to find the derivative of
\begin{align}
f = \textrm{tr} \{BA^+\} + \textrm{tr} \{B(A^+)^TCA^+B^T\}
\end{align}
with respect to $A$, where $A^+ …
- 291
3
votes
0
answers
83
views
Invexity of the $L_2$ norm
I have the following function:
$ f({\bf A,b}) = \| {\bf y - XAb} \|_2^2$
where ${\bf y}_{n \times 1}$ and ${\bf X}_{n \times p}$ are fixed, and ${\bf A}_{p \times r}$ and ${\bf b}_{r,1}$ are the varia …
- 291
3
votes
0
answers
127
views
Reducing $\ell_1$ norm of non-full-rank matrices
I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). … Can I build two new matrices ${\bf{X}}'_{p\times (r-1)}$ and ${\bf{Y'}}_{(r-1)\times q}$ such that the following three conditions hold:
$\bf{X}' \bf{Y}' = \bf{X} \bf{Y}$
$\|\bf{X}'\|_1 \leq \|\bf{X}\ …
- 291
2
votes
2
answers
1k
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Uniqueness of solution of a nonconvex optimization problem
What conditions need to be hold for a nonconvex optimization problem to have a unique solution?
Specifically, I have the following minimization problem that I'd like to know whether it has a unique s …
- 291
1
vote
1
answer
311
views
Modified Orthonormal Procrustes Problem
In the general orthonormal Procrustes problem, we want to find an orthonormal matrix $C$ to minimize $\|Y-XC\|_F^2$, where $Y$ is a known $n\times q$ matrix, $X$ is a known $n \times m$ matrix, and $C …
- 291
0
votes
2
answers
3k
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Convexity of the Frobenius norm of the product of two matrices
I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}_{n \times p … If it is not convex, can I impose some extra constraints on any of these matrices to make $f$ convex? …
- 291