Questions tagged [matrices]
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.
96 questions from the last 365 days
4
votes
1
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208
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Non-increasing property of a norm-like function over matrices
Let $P,Q$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
1
vote
1
answer
227
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Question on monotonicity of a norm-like function for matrices
Let $P_1,P_2$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
1
vote
1
answer
241
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Eigenvalues of analytic matrix families
Krešimir Veselić [[email protected]] asks:
Given the matrix family
$$ A(\varepsilon) = \begin{bmatrix} 0 & \Omega \\ −\Omega & −D-\varepsilon D′\end{bmatrix} $$
with $\Omega$...
- 47
4
votes
1
answer
173
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Question on power-nonnegative matrix
Let $P_1,P_2\in M_n(\mathbb R)$ be two orthogonal projections, i.e., $P_1^2=P_1=P_1', P_2^2=P_2=P_2'$ and assume that they are unitarily similar.
Let
$$
A=(P_1P_2)\circ(P_2P_1),
$$
where $\circ$ ...
- 857
3
votes
0
answers
76
views
Eigenvalues and characteristic polynomials of almost-principal minors
Discussion continued and significantly inspired from this MSE question.
Suppose we have computed through the Faddeev-Leverrier algorithm the characteristic polynomial of a matrix $A$: that is, not ...
- 644
1
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0
answers
66
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Classification of nilpotent matrices up to orthogonal similarity
Call a matrix $M$ over a field $K$ orthogonal if $M M^T=id$.
Question 1: Is there a classification of nilpotent matrices up to orthogonal similarity (at least for certain fields K), where two ...
- 28.5k
4
votes
1
answer
200
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The 'discriminant' of a matrix over a finite field
Let $\mathbb{F}_q$ be a finite field. Let $V=\mathbb{F}_q^m$.
Let $L_1,\dots, L_n$ be the one-dimensional subspaces of $V$, where $n=(q^m-1)/(q-1)$. Let $0\neq e_i\in L_i$ be non-zero representatives, ...
- 2,603
0
votes
0
answers
29
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Explicit orthogonal matrix with prescribed diagonal for the path graph Laplacian
Let $N \ge 2$ and consider the diagonal matrices
$$
\mathbf{D} = \operatorname{diag}(\lambda_0,\lambda_1,\dots,\lambda_{N-1}), \qquad
\mathbf{D}_0 = \operatorname{diag}\!\left(0,\frac{4}{N},\frac{8}{...
- 4,190
5
votes
0
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112
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Product of a matrix with the centralizer of another one
Let $A,B\in\mathrm{GL}_n(\mathbb{C})$ be two invertible matrices. Let $C(A)$ denote the centralizer of $A$ in $\mathrm{GL}_n(\mathbb{C})$. I wonder if the following is true.
Conjecture If $A$ and $B$ ...
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19
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5
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For which $n$ is the minimum max entry of an $n$ by $n$ orthogonal matrix known? Is $n=3$ already an open problem?
Crossposted on Mathematics SE, where the question Orthogonal matrices with small entries was brought to my attention, though it is about bounds rather than exact values.
Let $\| A \|_{\max} := \max\...
- 4,887
0
votes
0
answers
21
views
Existence of unimodular minor in expander graph bi-adjacency matrix
I am currently learning about expander graphs and expander codes from these notes by Venkatesan Guruswami. Given an $(n,m,D,\gamma,D(1−\epsilon))$ expander graph $G=(L \cup R, E)$ ($0 \leq \epsilon &...
- 111
1
vote
0
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187
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Hadamard determinant problem and the special orthogonal group
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PL{PL}$I was trying to reduce the Hadamard problem of calculating the maximum value of the determinant of a $\{1,-1\}$-matrix to the problem of ...
- 401
3
votes
1
answer
147
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Finite-index subgroups of Siegel parabolic subgroup
Consider the subgroup of the symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ consisting of matrices of the form $$\left\{ \begin{bmatrix} A & B \\ 0 & (A^t)^{-1} \end{bmatrix} \; | \; A ...
- 365
13
votes
2
answers
447
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Determinant of Hessenberg matrix with $1/n^2$ entries
Background
Consider the $(n \times n)$ Hessenberg matrix
$$ A_{n} := \begin{pmatrix} 1/2 & 1/3 & 1/4 & 1/5 & \dots & \dots & 1/(n+1) & \dots \\
...
- 5,387
6
votes
1
answer
261
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System of equations with an orthogonal matrix
I have six unit vectors $a_k$ and $b_k$ for $k \in \{1,2,3\}$. These are randomly drawn and are of dimension $3\times 1$.
Let $Q$ be an orthogonal matrix. I have noted that there are always$^\color{...
