**2**

votes

**1**answer

125 views

### Eigenvalues of a matrix constructed with simple logic

If a matrix can be constructed with simple bit-logic operations, is it also possible to find Eigenvalues with logic?
First I'll just say that my knowledge of logic is pretty much limited to ...

**2**

votes

**2**answers

227 views

### Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...

**4**

votes

**1**answer

463 views

### A question on Grassmannian

Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...

**2**

votes

**1**answer

284 views

### A submanifold of the space positive definite matrices

Consider the space of $n \times n$ positive definite symmetric matrices and let $\Sigma$ be one such matrix. We make this space into a Riemannian manifold $M$ by means of the metric ...

**7**

votes

**1**answer

476 views

### A question on eigenvalues

Let $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $A_{5}$ be linearly independent Hermitian matrices in the the space of $6$ by $6$ Hermitian matrices as a vector space over $\mathbb{R}$. Does there always ...

**1**

vote

**1**answer

125 views

### Weighted Spectral l-2 norms arising from matrix inner products

The spectral $l^2$ norm of a complex matrix is given by:
$\|A\|= \left( \sum_{k=0}^{N-1} s_k(A)^2 \right)^{1/2}$ where $s_k(A)$ are the singular values of $A$ ordered so as to be non decreasing in ...

**2**

votes

**2**answers

143 views

### Probability for a random positive-semidefinite matrix to not be positive-definite?

If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite ...

**3**

votes

**2**answers

208 views

### Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?

Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...

**7**

votes

**1**answer

695 views

### Frobenius-Perron eigenvalue and eigenvector of sum of two matrices

Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...

**3**

votes

**1**answer

87 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**5**

votes

**1**answer

167 views

### For a set of matrices $S$, find $X$ such that the elements of $SX$ commute

Let $S := \{A_0, A_1, \dots, A_d\}$, where $A_k \in \mathbb{C}^{n \times n}$, be a set of (generally noncommuting) matrices. I am interested in finding a nonsingular $X \in \mathbb{C}^{n \times n}$ ...

**2**

votes

**1**answer

83 views

### Updating $LU$ decomposition after adding a sparse matrix

How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...

**2**

votes

**0**answers

105 views

### A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...

**2**

votes

**1**answer

131 views

### Algorithmic Version of John's Decomposition of Convex Body

While reading textbooks on convex geometry, I heard about Fritz John's theorem on convex body, which is known as John's Theorem or John's Decomposition.
(I know that there are many variants, but this ...

**0**

votes

**0**answers

226 views

### Matrix inversion lemma and pseudoinverse?

Starting with this form of the matrix inversion lemma
$$
(A^T B^{-1} A + D^{-1})^{-1} A^T B^{-1} = D A^T (B + A D A^T)^{-1}
$$
substitute $B \rightarrow I$, $D \rightarrow \alpha^{-1} I$,
$$
...

**4**

votes

**3**answers

223 views

### Sherman-Morrison type formula for Moore-Penrose Pseudoinverse

Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the ...

**0**

votes

**0**answers

222 views

### expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...

**5**

votes

**3**answers

339 views

### Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R ...

**2**

votes

**1**answer

93 views

### Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...

**2**

votes

**1**answer

146 views

### What is the intuition behind Kontsevich-Iyudu-Shkarin result?

According to this paper, a cube of a composition of matrix inverse, matrix elements' inverse, and matrix transposition modifies 3x3 matrix by multiplying on left and right side by diagonal matrices.
...

**16**

votes

**5**answers

736 views

### Cayley-Hamilton revisited

Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let ...

**1**

vote

**0**answers

48 views

### Do GE rings have matrix completion?

If $R$ is a ring, $E_n(R)$ is the subgroup of the group $GL_n(R)$ generated by matrices obtained from the multiplicative identity matrix by replacing an off-diagonal entry by some $r \in R$. The ...

**6**

votes

**2**answers

245 views

### Powers of singular matrices and pairs of identical rows

Let $A$ be a square real or complex matrix. We’ll call $A$ special if among its rows (or among its columns) there are two identical ones, different from the zero vector, (Added:) and if it has no ...

**1**

vote

**0**answers

266 views

### Update the inverse of sum of two symmetric matrices

There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a ...

**8**

votes

**3**answers

1k views

### How to solve this quadratic matrix equation?

I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...

**2**

votes

**0**answers

50 views

### Rank of the smallest rank-reducing perturbation of a general matrix

Let $A$ be a general $m \times n$ matrix (not necessarily square, and not necessarily of full rank). Let $\| \cdot \|$ be a norm for $m \times n$ matrices that is induced from a norm for $m$ vectors ...

**4**

votes

**0**answers

103 views

### Concept of eigenvector restricted to nonnegative entries

Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...

**3**

votes

**1**answer

172 views

### Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\
...

**3**

votes

**2**answers

264 views

### Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements

is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :
\begin{pmatrix}
1 & b & 0 & ... & 0 \\\
b & 2 ...

**1**

vote

**1**answer

90 views

### How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?

The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian ...

**8**

votes

**1**answer

503 views

### $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?

I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...

**2**

votes

**1**answer

150 views

### When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...

**14**

votes

**4**answers

929 views

### Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so ...

**0**

votes

**0**answers

136 views

### What matrix operation is this?

Does this matrix operation have a name?
Let $A$ and $B$ be two $2^m$ by $2^n$ binary matrices. Then the element in the uth row and wth column of the new matrix is
$
\sum_{v=0}^{2^n-1} a_{u, v ...

**2**

votes

**1**answer

274 views

### Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend.
I am puzzled by the three necessary and sufficient conditions given there.
Could you provide me with some concrete ...

**7**

votes

**2**answers

185 views

### Sum of Difference of anti-diagonal matrix elements

Let $A \in \mathbb{R}^{n \times n}$, with elements $a_{ij}$
What conditions on $A$ are required for the following to be true?
There exists some vector $x \in \mathbb{R}^n_+$, $x \neq 0$ such that ...

**3**

votes

**1**answer

118 views

### On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...

**2**

votes

**0**answers

82 views

### integral stable conjugacy classes

Let $G$ be a semisimple simply connected group over $k$ algebraically closed field .
Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$.
We assume that ...

**4**

votes

**0**answers

163 views

### How to find the unitary matrices in this exponential matrix representation

In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and ...

**5**

votes

**0**answers

195 views

### Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for ...

**2**

votes

**1**answer

125 views

### Number rank-k 0-1 matrices (characteristic 0)

What is the number of $n\times n$ 0/1-matrices with rank $k$?
(The rank is taken over the rationals.)

**4**

votes

**1**answer

193 views

### Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)

Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...

**-2**

votes

**1**answer

304 views

### For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if ...

**4**

votes

**0**answers

101 views

### Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...

**2**

votes

**0**answers

140 views

### Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...

**2**

votes

**3**answers

175 views

### On matrices in linear forms with vanishing determinant

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...

**12**

votes

**1**answer

367 views

### Does the Hasse principle hold for the square root problem on symmetric matrices?

Let $S$ be an $n \times n$ symmetric matrix with rational entries.
It is known that the equation $XX^T=S$ has a solution in $\text{M}_n(\mathbb{Q})$
if and only if it has a solution in ...

**4**

votes

**1**answer

290 views

### Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...

**3**

votes

**0**answers

82 views

### Large prime divisors in entries of matrix powers

Are any examples known of an integer matrix $A$, such that the largest prime divisor of some specified entry of $A^n$ grows exponentially in $n$? How about where it just grows strictly faster than any ...

**0**

votes

**0**answers

98 views

### a very elementary question on the conjugated matrices

Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero .
We suppose that they have the same ...