**5**

votes

**2**answers

161 views

### The coefficient of a specific monomial in the expansion of the following polynomial

Let $a_{n,k}$ be the coefficient of $$X_1^{\frac{k(n-1)}{2}}X_2^{\frac{k(n-1)}{2}}\cdots X_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits_{1\leq i<j\leq ...

**3**

votes

**1**answer

42 views

### Complexity class of matrix generalization of knapsack problem

Let $n$ be a natural number, $u_+,v_+,u_-,v_-$ be real or complex column vectors of length $n$, and $M_1,M_2,\ldots,M_k$ be a finite collection of $n\times n$ real or complex matrices.
Consider the ...

**13**

votes

**2**answers

449 views

### A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**0**

votes

**1**answer

71 views

### a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} ...

**0**

votes

**0**answers

44 views

### Solving a system of quadratic equations over a subspace

Let $A \subseteq \mathbb{R}^{n}$ be subspace of dimension $d$, parametrized as $A=\{x|Vx=0\}$, where $V$ is a suitable $d \times n$ matrix.
Now a system of $m$ quadratic equations ...

**1**

vote

**1**answer

62 views

### Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...

**1**

vote

**0**answers

40 views

### Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...

**4**

votes

**1**answer

144 views

### Ideals in exterior algebras over the field with two elements

Suppose we have an exterior algebra over $\mathbb{F_2}$, say $R = \Lambda_{\mathbb{F_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}_2$ vectorspace. Let $x_1,\ldots,x_n$ be a basis of that ...

**3**

votes

**1**answer

132 views

### Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...

**1**

vote

**1**answer

31 views

### Similarity transform of a diagonalizable matrix that minimizes the Euclidean condition number

If I have a diagonalizable matrix $A = V\Lambda V^{-1}$, is there a way to show that for any similar $B$ such that $B = T\Lambda T^{-1}$, the Euclidean condition number $\kappa_2(B) \geq ...

**0**

votes

**1**answer

265 views

### How do Schubert classes form a basis for $H^{*}(Gr(k, n))$?

I've gone through many texts in algebraic geometry, specifically, Schubert calculus. They all claim that the Schubert classes $[\Omega_{\lambda}]$ form a basis for the cohomology ring of the complex ...

**2**

votes

**0**answers

410 views

### On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) \subseteq TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. Let $\bar{J} : TTM \longrightarrow ...

**3**

votes

**2**answers

264 views

### A question on determinant of a matrix polynomial

Let
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.
${\rm{P(}}\lambda {\rm{) = ...

**34**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**19**

votes

**0**answers

570 views

### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...

**4**

votes

**0**answers

106 views

### Hodge duality and the determinant of the product of two matrices

I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...

**2**

votes

**0**answers

51 views

### Determining Inconsistency of (first-order) Non-linear System of Equations [closed]

Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...

**2**

votes

**0**answers

62 views

### When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

It would be soo awesome if you could help!
For $ p \in \mathbb{N}$ consider the following $\mathcal{S(H)}^{p\times p}$-matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p}$ of nuclear operators ...

**0**

votes

**0**answers

24 views

### On a certain notion of sequence rank

Given a pair of length $n$ vectors $(a,b)$ denote $a\oplus b$ to be the operation that yields an $n\times n$ matrix with entry $i,j$ given by $a_i+b_j$ and every diagonal entry $0$
Given a zero ...

**2**

votes

**1**answer

114 views

### A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in ...

**2**

votes

**0**answers

130 views

### On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
...

**0**

votes

**1**answer

97 views

### Fixed point of quantum operations

A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...

**1**

vote

**0**answers

68 views

### Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix:
$C=AB$
where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...

**0**

votes

**0**answers

58 views

### Inverse Laplace transform of matrix exponential

I have the following Laplace-transformed, matrix-valued function:
$$U(s) = e^{As + B},$$
where $A$ and $B$ are diagonalizable, noncommuting (but very close to commuting, if that's useful -- $B$ is ...

**7**

votes

**3**answers

189 views

### Is there a standard name for the following type of linear operator?

Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...

**5**

votes

**5**answers

565 views

### Elementary linear algebra over a (possibly skew) field $K$

I have a number of questions which seem linked to me, about basic (?) linear algebra:
Given a field (possibly skew) $K$, and an superfield $L$, one can do linear matrix algebra with coefficients in ...

**3**

votes

**1**answer

90 views

### Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...

**9**

votes

**2**answers

248 views

### Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...

**1**

vote

**1**answer

62 views

### Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...

**4**

votes

**0**answers

55 views

### Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that ...

**0**

votes

**0**answers

44 views

### Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem :
$ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $
where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...

**1**

vote

**1**answer

64 views

### Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions.
https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors
Is ...

**3**

votes

**0**answers

70 views

### Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions:
All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix);
all principal minors are $>1$, ...

**12**

votes

**4**answers

515 views

### List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...

**-1**

votes

**1**answer

60 views

### Computing the inverse of a Cholesky decomposition [closed]

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...

**5**

votes

**2**answers

285 views

### Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = ...

**8**

votes

**2**answers

235 views

### Sets of matrices which are irreducible but not strongly irreducible

A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger ...

**2**

votes

**1**answer

191 views

### Number of Plücker relations for a Grassmannian

Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: ...

**2**

votes

**1**answer

103 views

### Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...

**1**

vote

**0**answers

96 views

### Boundary of pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**1**

vote

**1**answer

51 views

### Pragmatic Test for Total Unimodularity

I want perform a simple check for total unimodularity.
Question:
what, if anything, can be concluded from the fact, that $$det(A)=1,\ a_{ij}\in\{-1,0,+1\}\ \wedge\ a_{ij}^{-1}\in\{-1,0,+1\}$$
...

**2**

votes

**0**answers

140 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n ...

**2**

votes

**2**answers

142 views

### A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...

**3**

votes

**0**answers

86 views

### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
...

**2**

votes

**0**answers

74 views

### Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?

**0**

votes

**1**answer

50 views

### Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...

**2**

votes

**1**answer

67 views

### Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix ...

**4**

votes

**2**answers

160 views

### Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...

**6**

votes

**0**answers

180 views

### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...

**1**

vote

**1**answer

149 views

### Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$.
Here is a proof of this fact:
Proof: choosing a basis of ...