**5**

votes

**0**answers

205 views

### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...

**10**

votes

**0**answers

239 views

### Determinant inequality involving Hermitian, positive definite matrices

Question:
Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$
Show that:
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question ...

**0**

votes

**1**answer

88 views

### Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...

**0**

votes

**0**answers

93 views

### determinant inequality for symmetric positive definite matrices

Assume that A and B are symmetric, positive definite matrices of the same size.
For which set of real parameters $\alpha $ and $\beta$ the following relation holds
...

**3**

votes

**1**answer

124 views

### Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...

**19**

votes

**1**answer

829 views

### Recognize this strange expression from linear algebra?

I've come across an odd-looking expression and oh how I wish I had a more elegant description of it. Maybe someone who enjoys symmetric bilinear forms in characteristic two will recognize it? Or ...

**0**

votes

**2**answers

233 views

### Proof for a Rank-One Decomposition Theorem of Positive (semi) Definite Matrices

Consider the following result which I recently came across in a research paper in my area (Signal Processing)
Let $X$ be a $N\times N$ positive semidefinite (psd) matrix whose rank
is $r$. Let ...

**0**

votes

**1**answer

247 views

### Largest eigenvalue of the sum of hermitian matricies [closed]

Is there an expression for the largest eigenvalue of the sum of two hermitian matricies in terms of the spectrum of the same matricies?

**0**

votes

**1**answer

68 views

### Schur norm for self-adjoint operators

If $A$ is a $n \times n$ complex matrix then the Schur norm of $A$ is given by $$ || A||_S := \max_{||B||=1} ||A*B||,$$ where $||. ||$ is the operator norm and $*$ is the Hadamard (entry-wise) ...

**4**

votes

**1**answer

68 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

**5**

votes

**0**answers

123 views

### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...

**0**

votes

**1**answer

124 views

### Dimension of Commutator Space [closed]

For each $n\times n$ matrix $A$ with real entries the set
$$C(A)=\{X\in M_n(\mathbb{R}): AX=XA\}$$
is obviously a linear subspace of $M_n(\mathbb{R})$.
Can we recognize the dimension of this ...

**0**

votes

**1**answer

154 views

### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} ...

**2**

votes

**0**answers

75 views

### Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent
\begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation}
of a 3-tensor $W_{j,k,l}$ of ...

**0**

votes

**1**answer

69 views

### Adjusting matrix in generalized eigenvalue problem for the design of eigenfunctions

Take two matrices $A,B \in \mathbb{R}^{n\times n}$. I am considering the generalized eigenvalue problem ($\gamma \in \mathbb{R}, \phi \in \mathbb{R}^n$)
$$A\phi = \gamma B\phi.$$
Is there a ...

**7**

votes

**1**answer

305 views

### Block Matrix determinant

Consider the $k \times k$ block matrix:
$ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...

**2**

votes

**1**answer

103 views

### How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$
$k(x, x') = (1 - 2|x-x'|)$
is a positive semidefinite function?
It turns out to be psd function when ...

**3**

votes

**1**answer

134 views

### $\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to ...

**2**

votes

**0**answers

76 views

### Separating duality for TVS?

What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...

**0**

votes

**0**answers

67 views

### Bounding multiplications of PSD random matrices

Consider the following setup,
$(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$.
The ...

**9**

votes

**2**answers

362 views

### Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?

**4**

votes

**1**answer

378 views

### Who defined and who coined “module”?

The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been ...

**4**

votes

**1**answer

56 views

### Eigendecomposition of analytic Hermitian matrix-valued functions of several variables

If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...

**2**

votes

**1**answer

386 views

### Operator norm vs spectral radius for positive matrices

I believe the following statement should be true but somehow I don't see an argument:
For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...

**2**

votes

**1**answer

74 views

### Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1.
I am looking for the linear combinations of the columns of the matrix that sum to 0.
For example C1 + C2 - C3 = 0.
I want to find the linear ...

**3**

votes

**0**answers

88 views

### Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations
$C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$
The pair $(A,R)$ is referred to as a double description (DD) pair of the ...

**3**

votes

**1**answer

129 views

### Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers?
In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...

**4**

votes

**2**answers

214 views

### Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint").
I know that all the eigenvalues of matrices in this ...

**0**

votes

**0**answers

8 views

### Complex parameters in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...

**0**

votes

**1**answer

77 views

### Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...

**8**

votes

**1**answer

483 views

### Reference request: Book of Linear algebra from categorical point of view

Is there any book of Linear algebra in the modern language of Category theory?
I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...

**5**

votes

**2**answers

131 views

### When does a cone contain its dual cone?

Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...

**0**

votes

**1**answer

71 views

### Efficient way to find SVD of sum of projection matrices?

Lets say that we have n matrices of data $X_i : i \in [1, n]$. All $X_i$ have the same number of rows.
Their associated projection matrices are $P_i = X_i(X_i^T X_i)^{-1}X_i^T$
Also say that we have ...

**4**

votes

**1**answer

370 views

### Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?

Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...

**2**

votes

**1**answer

465 views

### An inequality involving traces and matrix inversions

The following question kept me wondering for some time:
Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...

**1**

vote

**1**answer

112 views

### Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let
\begin{equation*}
p = p\left(\kappa,d\right) := ...

**6**

votes

**2**answers

233 views

### Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group:
$$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$
Here, $H$ is an NxN skew-Hermitian matrix (for very ...

**1**

vote

**0**answers

36 views

### Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...

**2**

votes

**1**answer

76 views

### Is first term of my cost function convex?

I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...

**0**

votes

**1**answer

74 views

### Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*}
||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...

**3**

votes

**1**answer

140 views

### Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...

**3**

votes

**1**answer

94 views

### On the solution of a generalized Lyapunov equation

We shall reconsider the following equation
$$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$
where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix.
I met with this ...

**3**

votes

**1**answer

109 views

### weak version of a Baez-Crans 2-vector space?

Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...

**0**

votes

**0**answers

76 views

### Relation between the block maximum norm and the Euclidean norm of a matrix

I am trying to give answer to the following question:
Let's define de block maximum norm of a $N*M \times N*M$ matrix as
\begin{eqnarray*}
\parallel A \parallel_b = max_{x \neq 0} [ \parallel A x ...

**0**

votes

**0**answers

102 views

### Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...

**5**

votes

**1**answer

453 views

### Die hard nilpotent spaces

Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is nilpotent if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned ...

**2**

votes

**2**answers

157 views

### Matrix, singular values, Moore-Penrose-pseudoinverse

If A is any real mxn-matrix consider the block matrix
$\begin{pmatrix} E&A^T \\ A&0\end{pmatrix}$. This matrix seems to have close connections with pseudo inverse, svd etc. which are probably ...

**0**

votes

**0**answers

50 views

### Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...

**0**

votes

**1**answer

66 views

### Integral over Kronecker product

Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that
$$ \int_{0}^T A(t) dt $$ is invertible.
Does it then follow that the matrix
$$ \int_{0}^T A(t) \otimes ...

**4**

votes

**1**answer

78 views

### Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...