**0**

votes

**1**answer

106 views

### Parametrization of real diagonalizable matrices with given eigenvalues

Complex diagonalizable matrices with given eigenvalues can be conveniently parametrized as $A=T^{-1} \Lambda T$, where $T$ is any invertible matrix, and $\Lambda=diag(\lambda_1,...,\lambda_N)$ with ...

**12**

votes

**2**answers

233 views

### Order-preserving operator norms

Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell_p$ space; that is, we consider them as linear operators $\ell_p^n\to \ell_p^n$. When $p=2$, $M_n$ is a C*-algebra and ...

**3**

votes

**1**answer

203 views

### Derivative of trace of pseudo inverse

Given three matrices $A$ (broad), $B$ and $C$, I'd like to find the derivative of
\begin{align}
f = \textrm{tr} \{BA^+\} + \textrm{tr} \{B(A^+)^TCA^+B^T\}
\end{align}
with respect to $A$, where ...

**-2**

votes

**1**answer

96 views

### Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...

**5**

votes

**1**answer

254 views

### Grassmann-Plücker relations for permanents

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of ...

**1**

vote

**0**answers

79 views

### Dominant eigenvalue of sum of tridiagonal and diagonal matrices

Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...

**5**

votes

**1**answer

255 views

### What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$?
EDIT: Let's assume that $B$ ...

**0**

votes

**1**answer

77 views

### Eigendecomposition of a summation of matrices [duplicate]

Can anyone tell me if there's a way to relate the eigendecomposition of the result of a summation of matrices with the eigendecomposition of those matrices?
More specifically:
If I have a matrix $K = ...

**6**

votes

**1**answer

282 views

### Diagonalization for sums of Hermitian matrices

I found an interesting question about diagonalizable matrices,
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$.
Do there exist complex numbers $u\neq v$, such that $A+uB$ and ...

**2**

votes

**1**answer

153 views

### Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here:
Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...

**4**

votes

**2**answers

247 views

### Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...

**3**

votes

**2**answers

194 views

### Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody.
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...

**23**

votes

**1**answer

651 views

### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

**0**

votes

**0**answers

40 views

### Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem
\begin{align}
\lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\
&r_i\geq ...

**4**

votes

**1**answer

123 views

### Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that
$\|A\| < x$ and ...

**1**

vote

**0**answers

243 views

### Trace of Inverse matrix from Cholesky

This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case.
I have the matrix $\Sigma=LL^T$. Is there ...

**3**

votes

**1**answer

195 views

### About partial uniqueness of SVD

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let ...

**5**

votes

**0**answers

261 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**2**

votes

**0**answers

157 views

### Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...

**3**

votes

**1**answer

118 views

### submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...

**5**

votes

**1**answer

168 views

### An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...

**7**

votes

**0**answers

180 views

### Solving $P=AB,Q=BA$, in the unknowns $A,B$

Let $p\geq q$ $P\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. We seek $A\in M_{p,q},B\in M_{q,p}$ s.t. $P=AB,Q=BA$. The NS conditions for the existence of $(A,B)$ are given in
On the matrices AB and BA. ...

**7**

votes

**1**answer

520 views

### Is this Hankel matrix in trace class

Let A be the infinite Hankel matrix with the coefficient
$$A_{kj}=e^{(-t(k+j)^2)}-e^{(-t(k+j+2)^2)},$$ with $t$ a nonnegative real number.
Is $A$ in trace class with a norm bounded by an absolute ...

**0**

votes

**1**answer

82 views

### Understanding the derivation of a ML-estimator (statistics)

I'm trying to understand the derivation of a ML-estimator and more specifically the rewriting of the covariance matrix $\Sigma$. In this rewriting, a lemma is used to show that:
$$
\tag{1} ...

**3**

votes

**2**answers

245 views

### Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$,
the sequence $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...

**-1**

votes

**1**answer

66 views

### Strassen's algorithm for higher order matrices [closed]

So I am clear with the basic Strassen's algorithm. For a regular 2*2 matrix, it will perform 7 multiplications instead of the conventional 8 for regular matrix multiplication algorithms. The seven ...

**1**

vote

**2**answers

200 views

### What is the name for the type of matrices?

Let $ K $ be a field. We can recursively define matrices as
$ M_{a} = (a)$ for any $ a\in K $ and
$$ M_{a_1, \cdots, a_{2^i}} =
\begin{pmatrix}
M_{a_1, \cdots, a_{2^{i-1}}} & M_{a_{2^{i-1} ...

**8**

votes

**1**answer

281 views

### An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have
$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...

**2**

votes

**1**answer

83 views

### Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
...

**2**

votes

**0**answers

95 views

### Stationary Distribution for Markov-like system?

Let
\begin{equation}
A=
\begin{pmatrix}
0 & a_{1,2} & a_{1,3} \\
a_{2,1} & 0 & a_{2,3} \\
a_{3,1} & a_{3,2} & 0
\end{pmatrix},
\end{equation}
\begin{equation}
B=
...

**4**

votes

**2**answers

222 views

### Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...

**1**

vote

**1**answer

330 views

### Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$

The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset ...

**0**

votes

**1**answer

92 views

### Conditional Distribution of Inverse Wishart [closed]

Suppose $\begin{bmatrix}
K_{11} K_{12}\\K_{12}^T K_{22}
\end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix}
\Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22}
\end{bmatrix}\right)$.
What is the ...

**3**

votes

**1**answer

107 views

### Trace of multiplied positive definite matrices

I have to compute $Tr(K^{-1}\Sigma)$ where both $K$ and $\Sigma$ are symmetric positive definite matrices.
Question is considering that I have computed the Cholesky, $L_1$ of $K$ previously, is there ...

**-2**

votes

**1**answer

149 views

### Solving a difficult equation for a variable?

I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...

**2**

votes

**0**answers

243 views

### Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where,
$J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$
$A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$.
$B \in ...

**0**

votes

**0**answers

58 views

### Eigenvalues of a “Half-Kronecker ” Product

The Problem:
Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other
2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given).
Now we can construct a new matrix ...

**2**

votes

**1**answer

287 views

### Matrix equation XAX=B where the solution must be diagonal [closed]

$$X_{solution}=\arg\min_X \|XAX-B|_F \quad\mathrm{subject\ to}$$
X is square and diagonal
A is square and positive semi-definite
B is square and positive semi-definite
Any pointers or relevant ...

**-1**

votes

**1**answer

113 views

### Determinant of a sum of two Hankel matrices [closed]

First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? No.
Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if ...

**3**

votes

**1**answer

109 views

### The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...

**10**

votes

**1**answer

449 views

### Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...

**14**

votes

**1**answer

488 views

### A matrix completion problem

In their paper, "Corners of Normal Matrices," R. Bhatia and M.D. Choi ask the following question: Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such that the block ...

**0**

votes

**0**answers

60 views

### Property of quasipositive matrices

I saw this theorem stated in a paper without proof and I have difficulty proving it.
If $A$ is an $n\times n$ matrix with non-negative off-diagonal entries, let $s(A)$ be the real eigenvalue such ...

**1**

vote

**2**answers

105 views

### Mixing Numerical Range and inner product

Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as
\begin{align}
...

**-3**

votes

**1**answer

111 views

### Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...

**4**

votes

**0**answers

89 views

### Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...

**3**

votes

**1**answer

124 views

### Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$.
Define the ...

**6**

votes

**2**answers

145 views

### How to write this result (successive Schur complements compose nicely)

There is an easy theorem in linear algebra that "successive Schur complementations compose nicely": for instance, let's say I have a matrix
$$
M_0=
\begin{bmatrix}
A & B & C \\ D & E & ...

**3**

votes

**1**answer

194 views

### When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on ...

**1**

vote

**0**answers

326 views

### Inverse Transpose of Jacobian Matrix

Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by
\begin{equation}
f(x)\approx ...