All Questions
608 questions
10
votes
1
answer
923
views
Conjugation between commutative subalgebras of a matrix algebra?
Let $K$ be an algebraically closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two isomorphic commutative subalgebras of $M_n(K)$, it is true that there ...
- 545
9
votes
3
answers
356
views
Spectrum of orthogonality graph (2)
The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent.
I am only interested when $4|n$, since otherwise $\Omega(n)$ is ...
- 421
8
votes
2
answers
640
views
$2$-norm of idempotent matrix
Suppose $n > 1$ is an integer. Let $P \in \mathbb C^{n \times n}$ be a matrix such that $P^2=P$ and $1\leqslant {\rm rank}(P)<n$. Prove that $\Vert P \Vert_2 = \Vert I - P \Vert_2$.
I have been ...
8
votes
1
answer
492
views
Coordinate free isomorphism between $d+1$-dimensional antisymmetric rank $2$ tensors and $d$-dimensional symmetric rank $2$ tensors
The space of $(d+1)$-dimensional antisymmetric matrices has the same dimension as the space of $d$-dimensional symmetric matrices, $\frac12d(d+1)$. There are isomorphisms between the two spaces, e.g. ...
- 245
8
votes
1
answer
321
views
"Almost Hankelized" numerical Vandermonde
One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
1&x_2&x_2^2&\dots&x_2^{n-1}\\
\ldots&\ldots&\...
- 43.2k
8
votes
1
answer
2k
views
Symplectic block-diagonalization of a real symmetric Hamiltonian matrix
Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...
- 315
8
votes
1
answer
2k
views
A spectral inequality for positive-definite matrices
Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \...
- 13.5k
7
votes
3
answers
621
views
Reference for partial Hadamard matrices
Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
- 2,075
7
votes
1
answer
251
views
Solving for a set of points from projections
Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
S = \{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let $v \in \mathbb R^n$ and consider the image set (not counting ...
user45183
7
votes
1
answer
5k
views
How much can a diagonal matrix change the eigenvalues of a symmetric matrix?
Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that ...
- 449
7
votes
1
answer
1k
views
Strategies for bounding the spectral norm of a tensor?
Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...
- 20.2k
7
votes
2
answers
734
views
Does a non-singular matrix have a large minor with disjoint rows and columns and full rank?
Say I have an $n$-by-$n$ non-singular matrix $A$ all of whose diagonal entries are $0$. We call an $m$-by-$m$ minor of $A$ good if its set $I$ of row indices and its set $J$ of column indices ($I,J\...
- 20.2k
7
votes
3
answers
221
views
What is special in dimension $2$ (When characterizing isometries using the cofactor matrix)?
Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \...
- 6,741
7
votes
0
answers
264
views
Bound on gap between least eigenvalues of $n \times n$ correlation matrix and of its $(n -1) \times (n-1)$ submatrices
The following problem is motivated by one of my research problems.
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an ...
- 141
6
votes
1
answer
4k
views
Complexity for solving linear equations?
What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$?
I am interested in ...
- 544
6
votes
1
answer
996
views
Is there a smooth polar decomposition for non-invertible matrices?
Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...
- 6,741
5
votes
2
answers
266
views
Orthogonal Cauchy-like matrix
Given a $n \times n$ real Cauchy like matrix $C$, i.e., for real vectors $r$, $s$, $x$, $y$
$$
C_{ij} = \frac{r_i s_j}{ x_i - y_j}
$$
Can a Cauchy-like $C$ be orthogonal, i.e., $C C^T = I$ for $n ...
- 909
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
5
votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
5
votes
2
answers
2k
views
How to calculate the square root of matrix $A+B$ perturbatively?
$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...
- 977
5
votes
1
answer
289
views
The spectral radius of a binary matrix - polynomial growth?
(This is a follow-up to The spectral radius of a binary matrix)
Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.
QUESTION. Is there a $\delta\in\bigl(0,\frac12\...
- 2,135
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
- 11.1k
5
votes
1
answer
637
views
"Orthogonal complement" in $\mathbb{Z}_q^n$
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w \...
- 53
4
votes
1
answer
728
views
matrix congruence and smith normal form
Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
- 145
4
votes
1
answer
8k
views
Detection of Redundant Constraints
Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...
Countably Infinite
4
votes
1
answer
183
views
Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank?
This is a generalized version of Does a non-singular matrix have a large minor with disjoint rows and columns and full rank?
Let $A$ be an $n$-by-$n$ antisymmetric matrix of rank $r\geq \epsilon n$. ...
- 20.2k
4
votes
3
answers
763
views
Non-affine, projective vector field on $\mathbb{R}^n$
I wanted recently to discuss with a fairly elementary mathematics class the kinds of self-maps of Euclidean space that carry triangles to triangles. Obviously linear maps do this, and it seemed just ...
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
- 41
4
votes
2
answers
217
views
Collection of projection operators in finite dimension and algebraic techinques
Consider a set of linearly independent vectors $\{x_1,\dots,x_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P_S$ be the (orthogonal) projection (operator) ...
- 1,731
3
votes
1
answer
3k
views
Singular value decomposition of random rectangular matrices
Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the ...
- 884
3
votes
1
answer
381
views
Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices
Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\geq 0$ then $\bigl(\|(a_{ij},b_{ij})\|...
- 175
3
votes
1
answer
370
views
The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$
What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
- 4,058
3
votes
2
answers
453
views
Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
3
votes
3
answers
2k
views
When does a Vandermonde-like matrix have full rank
I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form:
$X_{i}= [1, x_{i}, y_{i}, x_{i}^...
- 133
3
votes
1
answer
455
views
Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE
$$f(t)=\langle\exp(-...
- 2,242
2
votes
1
answer
342
views
When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?
Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...
2
votes
1
answer
150
views
Can the supremum of this quotient of spectral radii be reached?
Let $V$ be a finite dimensional complex inner product space. If $A_1,\dots,A_r\in L(V)$, then define a mapping $\Phi(A_1,\dots,A_r):L(V)\rightarrow L(V)$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\...
2
votes
0
answers
185
views
Infinite products from the fake Laver tables-Now with no set theory
We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have
$2^{n}*_{n}x=x$,
$x*_{n}1=x+1\mod 2^{n}$,...
2
votes
1
answer
1k
views
Derivative of eigenvectors of an Hermitian matrix
In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
- 23
2
votes
0
answers
372
views
What is the Birkhoff norm of a Perron vector?
Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...
- 6,962
2
votes
0
answers
92
views
Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices: follow-up
I asked the following question here: "Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\...
- 175
2
votes
0
answers
359
views
Convex combinations $A$ of $n\times n$ permutation matrices such that every entry in $A^{k}$ is $1/n$
Recall that a doubly stochastic matrix is a square matrix $A$ with non-negative entries such that the sum of each row is $1$ and the sum of each column is $1$. The Birkhoff-von Neumann theorem states ...
1
vote
1
answer
381
views
A geometric property of singular matrices
Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let $M=...
- 356
1
vote
1
answer
462
views
Is this a full rank matrix? [closed]
According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
- 356
1
vote
1
answer
223
views
Worpitzky-like identities?
Let $$r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$
Computations suggest that $$r_{2k}(x)=\sum_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r_{2k+1}(x)=\sum_{j=0}^{k^2-k}a(2k+1,j)\...
- 5,622
1
vote
1
answer
276
views
Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements
I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that
$$
A = \begin{bmatrix}
-\alpha & \alpha & -\gamma^{-1} & 0\\
\beta &...
- 207
11
votes
1
answer
627
views
One question on circulant $\pm1$ matrices
Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property
$$AA^T=(n-1)I+J$$
where $I$ is the $n \times n$ identity matrix and $J$ ...
- 696
115
votes
3
answers
5k
views
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
- 5,063
66
votes
2
answers
8k
views
Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
- 2,071
66
votes
3
answers
4k
views
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
- 6,162