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Results tagged with matrix-theory
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user 8430
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
13
votes
Accepted
A generalization of van der Waerden's conjecture
The conjecture is false. Here is a counterexample.
\begin{equation*}
A = \begin{bmatrix}
\tfrac18 & \tfrac4{15} & \tfrac1{10}\\
\tfrac18 & \tfrac4{15} & \tfrac1{10}\\
\tfrac68 & \tfrac7 …
- 2,493
7
votes
Accepted
What's the best orthonormal matrix to align two matrices in the operator norm sense?
The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational Math …
- 28.6k
5
votes
Accepted
Better name for “vec transposition permutation matrix”?
This matrix is known as the commutation matrix. For more details, please see Chapter 3, Section 7 of: Matrix differential calculus by Magnus and Neudecker.
- 28.6k
10
votes
Accepted
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for ...
Write $Y=SDS^{-1}$, where $D$ is diagonal (since $X_1$ and $X_2$ are psd, $Y$ is diagonalizable). Then, observe that
\begin{equation*}
f(p) = \text{tr}(S^{-1}(I+pSDS^{-1})^{-1}SS^{-1}(I-p^2SDS^{-1}) …
- 28.6k
6
votes
Accepted
On a trace condition for positive definite $2\times 2$ block matrices
For any unitarily invariant norm it can be shown that
\begin{equation*}
\|X\| = \left\Vert
\begin{bmatrix}
A & C\\
C^* & B
\end{bmatrix}
\right\Vert \le \|A\| + \|B\|.
\end{eq …
- 28.6k
7
votes
Parametrization of positive semidefinite matrices
This paper considers optimization problems on the set of low-rank PSD matrices, and in particular talks about operating in a quotient space to deal with the non-uniqueness.
See also this work that i …
- 28.6k
8
votes
Accepted
"Additive version" of Kronecker product
Federico already mentioned the keyword. The precise answer may be found among others as Theorem 13.16, of this book. (That theorem makes a restriction to real matrices, but that is not necessary).
- 28.6k
17
votes
The sum of squared logarithms conjecture
Lev's proof reminded me of two papers, and unless I'm doing something silly, the said conjecture follows as a corollary of those papers. The answer below is just meant to supplement Lev's result, and …
- 28.6k
5
votes
Accepted
On closest unitary matrix
It turns out the the previous answer had the right ingredients, but in the wrong combination. Here is a cleaner proof.
Notation: Let $s_j(X)$ denote the $j$-th singular value of a matrix $X$ (we assu …
- 28.6k
2
votes
On closest unitary matrix
EDIT The claim below is false (stupid error, as noted in the comments). I will leave the answer below as a potential approach; am checking if the idea can be fixed!
Here is a proof that holds under …
- 28.6k
7
votes
Accepted
On a determinant inequality of positive definite matrices
A quick counterexample to your conjecture is
\begin{equation*}
A = \begin{pmatrix}
13 & 3 & -13 & -5\\
3 & 4 & -3 & 4\\
-13 & -3 & 13 & 5\\
-5 & 4 & 5 & 10\\ …
- 28.6k
5
votes
Accepted
What is such an equation called?
Akin to my comment, this equation can be called a nonlinear generalized eigenvalue problem. Usually, $f$ and $g$ are polynomials in $\lambda$, but more general nonlinearities might be allowed. In gene …
- 28.6k
4
votes
S-matrix conjecture: status?
To my knowledge, this problem is still open. There has been partial progress on it in the past few years, but it still seems quite far from resolved.
The latest paper that I am aware of is: here, tho …
- 28.6k