**5**

votes

**1**answer

167 views

### Eigenvalues of $X$ in the metric of $Y$

What does this statement describe? $X$ and $Y$ are matrices.
The eigenvalues of $X$ in the metric of $Y$.
I've not seen this language used before in this fashion and I don't really know what ...

**0**

votes

**1**answer

51 views

### Is spectral properties a general term for condition number?

I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...

**0**

votes

**0**answers

108 views

### Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...

**2**

votes

**1**answer

88 views

### Eigenvalues of product of symplectic matrices

I have 2 symplectic matrices $X_{1},X_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X_{1} \cdot X_{2}$ is also symplectic.
Question: Are there any theorems which allow me to express eigenvalues of ...

**1**

vote

**2**answers

64 views

### Sensitivity analysis in minimum norm problems under a linear constraint

Suppose $\Delta$ is some nice topological space, say compact, and Hausdorff.
Let $A:\Delta \rightarrow \mathbb{R}^{m\times n}$ be a continuous $m\times n$ matrix valued map. Let $b\in \mathbb{R}^{m}$ ...

**0**

votes

**0**answers

40 views

### Can the maximal eigenvalue of Toeplitz hermitian be bound by one entry?

Let $T$ be $N \times N$ toeplitz hermitian matrix. Assuming we control the entry $T_{N,1}$ with the other entries fixed. Can we determine the maximal eigenvalue of $T$, or at least bound it?

**11**

votes

**1**answer

215 views

### Is this generalization of eigenvalue and eigenvector studied?

While thinking about what it means for observables to be simultaneously measurable in quantum mechanics I came up with the following concepts, which I will call "linearly indexed" versions of standard ...

**7**

votes

**1**answer

104 views

### $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...

**5**

votes

**1**answer

163 views

### Continuity of solutions to $Av=b$

Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of complex-valued, continuous functions on $X$. Let $A$ be a matrix with entries from $C(X)$ of size $m\times n$ and $b\in ...

**8**

votes

**1**answer

256 views

### Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?

How can I efficiently compute
$\mathrm{trace}(A(B^{-1}))$
where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of ...

**9**

votes

**1**answer

466 views

### What is the spin connection in 9 dimensions as opposed to 5 dimensions?

From Spin Connection in 5 dimensions I can define a massless fermion's covariant derivative on a curved manifold as
$$
\nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi
...

**3**

votes

**2**answers

271 views

### Is it always possible to “separate” the eigenvalues of an integer matrix?

Call a square matrix Galois-irreducible if all its eigenvalues are Galois conjugates of each other.
Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always ...

**1**

vote

**1**answer

200 views

### Testing $0$ for a determinant like function [closed]

Given $A\in\Bbb Z^{n\times n}$ we have $$Det(A)=\sum_{\sigma\in S_n}(-1)^{sgn(\sigma)}\prod_{j=1}^nA_{j\sigma(j)}$$
We can test when this is $0$ by looking at the rank in polynomial time.
Can either ...

**4**

votes

**3**answers

157 views

### Fast Upper Triangular Matrix Exponentiation

Let $Q_n$ be a $n\times n$ matrix with $Q_n=\begin{pmatrix} -\lambda_1-\mu_1 & \lambda_1 & 0 & \cdots\\ 0 & -\lambda_2-\mu_2 & \lambda_2 & \cdots\\ \vdots & \vdots & ...

**1**

vote

**0**answers

49 views

### References for a minor variant of the Rayleigh quotient

I believe this variant of the Rayleigh quotient inequality must be well known but I could not find references for it online. It's proof is straightforward.
Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...

**0**

votes

**1**answer

51 views

### Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let ...

**10**

votes

**1**answer

193 views

### On Sampling rank $r$ matrices

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $\{-d,\dots,-1,0,1\dots,d\}$ uniformly.
What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?
Is there a nice parametrization of ...

**6**

votes

**1**answer

281 views

### Encyclopedia of properties of nonnegative matrices

I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $A_{i,j}\geq 0$ for all $i,j$.
Chapter 8 in ...

**2**

votes

**2**answers

107 views

### Orthogonal Procrustes problem with Absolute Values

Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising
$$ \left\lVert \, |O| - T \right\lVert_F,$$
where $\lVert \cdot \lVert_F$ denotes the Frobenius norm, and $| \cdot |$ ...

**7**

votes

**1**answer

152 views

### approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...

**2**

votes

**2**answers

103 views

### On $XX'=I$ such that $AX=XB$ is true when $A,B\in\{0,1\}^{n\times n}$

Given real symmetric matrices $A,B\in\{0,1\}^{n\times n}$ is it true that $$AX=XB$$ has a solution of form $X$ a permutation matrix iff a solution with $XX'=I$ exists? We are over reals.
It is clear ...

**5**

votes

**1**answer

348 views

### Complexity of linear solvers vs matrix inversion

Solving linear equations can be reduced to a matrix-inversion problem, implying that the time complexity of the former problem is not greater than the time complexity of the latter. Conversely, given ...

**7**

votes

**1**answer

298 views

### Geometry of Hermitian rank $\leq r$ matrices

Let $M_n^{sa}$ be the space of $n\times n$ complex Hermitian matrices, let $r < n$, and let $E$ and $F$ be (real) linear subspaces of $M_n$ with ${\rm codim}(E) < r^2$ and ${\rm codim}(F) = 1$. ...

**7**

votes

**3**answers

182 views

### On $XX'=I$ such that $AX=XB$ is true

Given list of symmetric matrices $\{A_i,B_i\}_{i=1}^r\in\Bbb R^{n\times n}$ where $r\in\Bbb N$ is arbitrary what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ ...

**4**

votes

**0**answers

168 views

### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...

**4**

votes

**0**answers

47 views

### Isometry theorem, exists homeomorphism that carries each fiber isomorphically onto itself

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically onto ...

**5**

votes

**1**answer

152 views

### Sum of the absolute eigenvalues of A>=B

Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ ...

**3**

votes

**3**answers

142 views

### Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...

**2**

votes

**1**answer

86 views

### How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?

Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...

**6**

votes

**2**answers

395 views

### Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...

**1**

vote

**1**answer

248 views

### Decompose $\Lambda^3(V \otimes W)$ [closed]

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes ...

**3**

votes

**1**answer

114 views

### Uniqueness of the reduced rank QR decomposition

Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$.
I read about the rank reduced $QR$ decomposition (here for example) as the QR variant where $A=QR$ ...

**4**

votes

**2**answers

62 views

### Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := ...

**12**

votes

**0**answers

187 views

### How does a permutation $P$ affect the singular value $\sigma_{\text{max}}(Q^\top P^\top Q)$ for orthogonal $Q$?

Let $q_i$ for $i=1,\ldots,m$ be the columns of the matrix $Q\in\mathbb{R}^{n\times m}$, $n\geq 2m$, which are pairwise orthonormal ( i.e.
$q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...

**12**

votes

**2**answers

336 views

### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?
For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...

**3**

votes

**0**answers

72 views

### When is $\left[\begin{smallmatrix} D_1 & B \\\\ -B^T & D_2 \end{smallmatrix} \right]$ $\mathbb{R}$-diagonalizable?

Is there some block-wise characterization of $\mathbb{R}$-diagonalizability (by similarities) of
$$\begin{bmatrix} D_1 & B \\\\ -B^T & D_2 \end{bmatrix},$$
where $D_1$ and $D_2$ are real ...

**8**

votes

**0**answers

149 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...

**1**

vote

**1**answer

179 views

### Sum of Eigenvectors Entries of an Adjacency Matrix

I have a question regarding the sums $\sum_{i=1}^{n}v_{j}\left(i\right)$ where $v_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length.
Ordering the eigenvectors by ...

**9**

votes

**0**answers

158 views

### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here.
Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...

**1**

vote

**2**answers

260 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**8**

votes

**0**answers

89 views

### Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...

**4**

votes

**0**answers

137 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

**3**

votes

**0**answers

83 views

### Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the ...

**8**

votes

**2**answers

191 views

### generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix
the maps
$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$
x\longmapsto ...

**2**

votes

**0**answers

100 views

### Notions of singularity for symmetric bilinear maps

Given a symmetric, bilinear map $B : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$, a common notion of nondegeneracy is the following:
$$
\mbox{If } B(x,y) = 0 \mbox{ for all } y, \mbox{then } ...

**20**

votes

**1**answer

711 views

### Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that $0 ...

**7**

votes

**1**answer

404 views

### How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with ...

**3**

votes

**1**answer

168 views

### Smallest eigenvalue gap of a non-symmetric random matrix

The question: Let $A$ be the matrix whose each element is an independently generated random variable which is uniform on $[0,1]$. One can see that the eigenvalues of $A$ will be distinct almost ...

**8**

votes

**1**answer

192 views

### Pair of square matrices related by traces formulas

Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How ...

**3**

votes

**1**answer

171 views

### comparing norms of tensor product of two Hilbert spaces

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...