4 votes
Accepted

What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Consider the linear operator $S:X\mapsto AXA^T$, on the space of matrices $\mathbb R^{n\times n}$, with the spectral norm. Its operator norm is $\|A\|^2<1$ so $1-S$ is invertible, with inverse ...
Pietro Majer's user avatar
  • 56.1k
3 votes
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Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

No, there are counterexamples with $p=5$. Let me restate the question. For a square matrix $A$ over any field, write $Y(A)=\{M:\forall Z\in C(A):\operatorname{Tr}(MZ)=0\}.$. Your question is whether $...
YCor's user avatar
  • 60k
3 votes

Norm of a matrix function of a vector $x$

$\newcommand\R{\mathbb R}$As noted by Denis Serre and Christian Remling, the exponents $1/2$ and $3/2$ must be interchanged -- otherwise, $\|A(x)\|_2\to\infty$ if $x=tu$, $u$ is a fixed unit vector, ...
Iosif Pinelis's user avatar
1 vote

Right inverse of integer matrix

Using the Pinelis' and Steinberg's posts, we can -easily - deduce $\textbf{Proposition.}$ Let $m<n$, $A\in M_{m,n}(\mathbb{Z})$ and let $S$ be its Smith normal form. Then $A$ admits a right inverse ...
loup blanc's user avatar
  • 3,564
1 vote

Right inverse of integer matrix

If $A$ is $m\times n$, then for a right inverse to exist it is clearly necessary that the rank of $A$ be $m$ and hence $m\le n$. For a right inverse of $A$ with integral entries to exist it is ...
Iosif Pinelis's user avatar
1 vote

Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex

This equation cannot be solved in terms of the Lambert $W$ function or any other known functions -- even if a, r1, and r2 are real, let alone complex. Here is what Mathematica says about this:
Iosif Pinelis's user avatar
1 vote

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

$\DeclareMathOperator\spectrum{spectrum}\DeclareMathOperator\GL{GL}$As Nathaniel wrote, the case $X\in M_n(\mathbb{Q})$ is not difficult. Let $p>0$ be an integer s.t. $pX\in M_n(\mathbb{Z})$. The ...
loup blanc's user avatar
  • 3,564
1 vote

Are there any applications of the algebraic polar decomposition?

The "algebraic polar decomposition" was introduced pre-Kaplansky by Choudhury and Horn as a A Complex Orthogonal-Symmetric Analog of the Polar Decomposition (1987). An application to improve ...
Carlo Beenakker's user avatar

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