**0**

votes

**0**answers

9 views

### I cant solve this trace optimization prolem

Can you help me to solve this problem?
enter image description here

**0**

votes

**0**answers

22 views

### LU factorization for $I+A$ (A skew-symmetric)

The matrix $M=I+A\in \mathbb{R}^{n\times n}$, where $I$ is the identity and $A=-A^T$ is skew-symmetric satisfies
$$
Mx\cdot x = \|x\|^2 \quad\text{for all }x\in\mathbb{R}^n.
$$
Therefore, $M=LU$ has ...

**-2**

votes

**0**answers

29 views

### Systems of linear equations [on hold]

Consider the system of equations:
x + 2y - z = -3
3x + 5y + kz = -4
9x + (k+13y) + 6z = 9
(A) express these equations as an augmented matrix.
(B) show that this matrix can be reduced to:
[1 , 2 , ...

**-2**

votes

**0**answers

31 views

### Eigenvalues of block Circulant matrix [on hold]

Let us consider the matrix $A$ as follows
\begin{align}
A=\left(\begin{array}{cc}
B & C \\ -C & B
\end{array}\right),
\end{align}
where $B=\left(\begin{array}{ccc}
\alpha^R & \beta^R ...

**0**

votes

**0**answers

51 views

### Problems in Nonlinear Matrix Equation [Question1] [on hold]

Let $\theta, \beta$ be 3 × 3 skew-symmetric matrices, and $\sigma$ be a 3 × 3
matrix. Find symmetric matrices S and T such that
$$(S − \theta) · (T − \beta) = \sigma$$
The answer must provide a clear ...

**0**

votes

**0**answers

40 views

### Characterize matrices complying to certain constraints [on hold]

Characterize those matrices $X$ (real symmetric), $Y$ (real positive definite), $R$ (real diagonal) and $F$ (real diagonal) such that
$XRY + YRX = 0$, (1)
$YRY - XRX = F$. (2)
Can we give some ...

**1**

vote

**0**answers

12 views

### the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...

**1**

vote

**1**answer

116 views

### Generalising the cyclic property of the trace of a matrix

Are their other functions of a complex square matrix, not trivially related to trace, which also posses the cyclic property?
Furthermore, do all such functions $f(A)$ depend only on the spectrum of ...

**13**

votes

**1**answer

219 views

### A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...

**-1**

votes

**0**answers

15 views

### 2-norm of a canonical Jordan form and spectral radius [migrated]

Let J be a real square matrix which has a canonical real Jordan form. Is it true that the 2-norm of J is equal to its spectral radius?
P.S.:
The 2-norm of J is ...

**0**

votes

**1**answer

119 views

### Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$.
What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product
...

**0**

votes

**1**answer

131 views

### Positive Semidefinite matrix [closed]

Let $A$ be an $n\times n$ symmetrix matrix, if $\forall i$,
$a_{ii}\geq |a_{ij}|,\forall j$
satisfies, can we say that $A$ is a positive semidefinite matrix? I tried to find a counter example, but ...

**0**

votes

**0**answers

87 views

### Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$

Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix ...

**4**

votes

**1**answer

53 views

### An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:
$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$
Note that if $A=I$ then ...

**6**

votes

**1**answer

189 views

### On an inequality among determinants

For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite.
In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$,
$X\ge ...

**3**

votes

**1**answer

151 views

### Switching representatives of right coset in a paper on fundamental domain of tree of GL(2)

I have a question concerning the following paper: "The fundamental domain of the tree of GL(2) over the function field of an elliptic curve" by Shuzo Takahashi. (1993, Duke Math. J., Vol. 72, No. 1). ...

**0**

votes

**0**answers

46 views

### Prime Hadamard Matrices

Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, ...

**0**

votes

**0**answers

31 views

### Diagonalization of a Toeplitz matrix [migrated]

Let $0<\lambda\leq1$ so that the $n \times n$ matrix
$$\Sigma = \begin{pmatrix}
1&1-\lambda& \cdots &1-\lambda\\
1-\lambda&\ddots&\ddots& \vdots\\
\vdots ...

**2**

votes

**1**answer

81 views

### Space of matrices B for which there is a solution to Bx=c for a given c

Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$.
Is there some name for the set $\mathcal{B}_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ ...

**2**

votes

**0**answers

55 views

### the annihilator of cokernel in a particular case

Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...

**1**

vote

**1**answer

87 views

### Diagonalization of 4th order tensors

I have been wondering about the following problem...
Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...

**-2**

votes

**0**answers

23 views

### How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space? [migrated]

I have the equation of two 5th degree polynomials which they don't intersect with each other. Each curve is made of 100 points and these two curves look similar but there are small differences. I am ...

**5**

votes

**0**answers

279 views

### A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
...

**0**

votes

**1**answer

60 views

### Reference request for: inverse of a non-singular M-matrix has all elements non-negative?

Does anyone know the best (earliest?) reference please for the proof that the inverse of a non-singular M-matrix has all elements non-negative?

**0**

votes

**0**answers

31 views

### Nonlinear matrix equation (transpose) [duplicate]

Let $H$, $M$ and $N$ be 10 by 10 matrices over the integers. If $M$ and $N$ are known, how do you solve for $H$ from the following equation?
$M = H N H^T$
where $H^T$ is the transpose of H.

**2**

votes

**1**answer

77 views

### Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...

**5**

votes

**7**answers

457 views

### Source for roots of matrix polynomials?

A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...

**3**

votes

**0**answers

114 views

### Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!

**8**

votes

**1**answer

118 views

### A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...

**2**

votes

**0**answers

124 views

### Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...

**6**

votes

**2**answers

361 views

### Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...

**3**

votes

**0**answers

89 views

### Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...

**2**

votes

**1**answer

79 views

### dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...

**0**

votes

**0**answers

28 views

### represent hankel matrix by low rank tensorial approximation

suppose that we have given following matrix
\begin{matrix}
x_1 & x_2 & ..x_p \\
x_2 & x_3 & ...x_{p+1} \\
. & .& . & \\
x_{N-p+1} & x_{n-p+2} &... x_n
...

**0**

votes

**1**answer

160 views

### Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...

**0**

votes

**0**answers

101 views

### A matrix rank problem over finite fields: Is that a known problem?

I have already asked the same question on cstheory.SE, but I haven't got an acceptable answer. So, I decided to ask it here.
It might be a known problem, however.
Let $A \odot B$ denote elementwise ...

**1**

vote

**0**answers

65 views

### spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e.,
...

**0**

votes

**1**answer

92 views

### Simultaneous triangularizability over a commutative ring

Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if ...

**3**

votes

**2**answers

161 views

### is 1/max(i,j) a bounded matrix on hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$.
It would be sufficient to know if the Lehmer matrix ...

**1**

vote

**1**answer

44 views

### Maximising a Rayleigh quotient over a subspace II

Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form
$$\tag{1}
...

**1**

vote

**1**answer

64 views

### Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...

**4**

votes

**1**answer

160 views

### Vector Spaces of Symmetric Matrices of Low Rank

Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where ...

**5**

votes

**1**answer

230 views

### Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|_p = \sup_{x\neq ...

**0**

votes

**1**answer

68 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**0**

votes

**0**answers

36 views

### Spanning Hadamard product powers (Schur products) in Euclidean space

This is a question I asked last year over at stackexchange. Got no answer, it might be sufficiently non-trivial to justify bringing it into here.
Fix two $k$-vectors $\mathbf u$ and $\mathbf v$, and ...

**1**

vote

**0**answers

66 views

### Inverse of matrix of generalised harmonic numbers

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the ...

**10**

votes

**1**answer

244 views

### doubly-stochastic isomorphisms of graphs

A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a doubly-stochastic automorphism of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph ...

**6**

votes

**2**answers

239 views

### Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...

**6**

votes

**0**answers

65 views

### LU-factorization for strongly elliptic matrices

For symmetric and positive matrices, we know that the cholesky factorization is stable. For a general matrix $A\in\mathbb{R}^{n\times n}$ with
$$
Ax\cdot x \geq C_{ell}|x|^2\quad\text{for all }x\in ...

**1**

vote

**1**answer

74 views

### Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...