**0**

votes

**1**answer

39 views

### A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result:
For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...

**0**

votes

**0**answers

45 views

### Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...

**2**

votes

**1**answer

90 views

### Carathéodory's theorem for $SO(3)$?

Let $Q \in \operatorname{Conv} SO(3)$.
Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}},
R_{i} \in SO(3)$?
An approximation ...

**0**

votes

**0**answers

83 views

### Checking whether the determinant of a matrix is non-zero [on hold]

For a positive integer $n$, let $c$ be the number of ordered integer tripartitions $(a,b,n-a-b)$ of $n$. Let $F = ( f_{ij} )$ be a $c \times c$ matrix where $f_{ij} = (i+1)^{a + b} (i+2)^{b}$ where $i$...

**0**

votes

**0**answers

51 views

### Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$.
Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is
$M[i,j]={(...

**5**

votes

**0**answers

88 views

### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$ [on hold]

Does anybody know an algorithm to solve the following matrix equation?
$$X^{-1}=\sum_{i=1}^n D_i X A_i$$
where $D_i$s are diagonal and $A_i$s are symmetric matrices.
It would be great to have an ...

**0**

votes

**0**answers

32 views

### $nD$ rotation around a general $(n-2)$-dimensional subspace [closed]

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$
corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...

**-3**

votes

**0**answers

77 views

### Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [closed]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...

**2**

votes

**0**answers

55 views

### How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix
$$A(n) =
\begin{pmatrix}
B_{1} & B_{2} & \cdots & B_{n} & 1 \\
B_{n} & B_{1} & \cdots & B_{n-1} &...

**-1**

votes

**0**answers

62 views

### When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?

**4**

votes

**1**answer

72 views

### information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...

**0**

votes

**0**answers

20 views

### Number of Asymmetric, Balanced Permutation Matrices

let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the ...

**3**

votes

**0**answers

25 views

### Selecting columns from multiple matrices to form a well-conditioned matrix

Given multiple matrices of the same size, is there a way to select one column from each matrix to form a well-conditioned matrix?
For example, given four 4-by-10 matrices A, B, C, D (real, positive, ...

**0**

votes

**1**answer

81 views

### How does the rank of $C_i$ change with $i$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...

**0**

votes

**0**answers

13 views

### Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix).
With some ...

**0**

votes

**0**answers

45 views

### SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices
$$
A = E'E
$$
Practically this can be done easily using SVD ...

**1**

vote

**2**answers

61 views

### How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...

**0**

votes

**0**answers

34 views

### Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, $$\lvert\lambda_1-...

**0**

votes

**1**answer

63 views

### Matrix norm inequality for C*-Algebras [closed]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...

**1**

vote

**0**answers

39 views

### Matrix transformation [closed]

I want to show that
$(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis
with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$
is equivalent to $M_\gamma$ has no purely imaginary ...

**6**

votes

**2**answers

820 views

### What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...

**1**

vote

**0**answers

68 views

### Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...

**6**

votes

**1**answer

212 views

### A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...

**4**

votes

**1**answer

93 views

### Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...

**6**

votes

**1**answer

232 views

### coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...

**2**

votes

**1**answer

159 views

### The class of $(-1,0,1)$-matrix with all row sums and column sums equalling to $0$

Let $n$ be an even positive integer and $W_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ satisfying all row sums and column sums are equal to $0$.
For any $M\in ...

**5**

votes

**2**answers

291 views

### Explicit solution to a Rayleigh quotient equation

For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
\biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...

**4**

votes

**1**answer

75 views

### Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation
$$
A_p = A + \Delta
$$
is there a way to represent
$$
(A_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A_p)^{\star}$ ($(A)^{\star}...

**11**

votes

**3**answers

240 views

### Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

I've read that if $M_1, \dots, M_n$ are matrices in $\mathrm{SL}(2, \mathbb{Z})$ whose product is the identity, and each is conjugate to the shear
$$ \begin{pmatrix} 1 & 1 \\ 0 & 1\end{...

**0**

votes

**1**answer

86 views

### Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...

**3**

votes

**0**answers

85 views

### Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...

**0**

votes

**2**answers

59 views

### Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

**4**

votes

**1**answer

96 views

### I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?

This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated.
Question
I have a very ...

**3**

votes

**1**answer

136 views

### Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...

**1**

vote

**0**answers

12 views

### non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...

**9**

votes

**0**answers

206 views

### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...

**1**

vote

**1**answer

84 views

### Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that $A=QBQ^{...

**-2**

votes

**1**answer

106 views

### Eigenvalues of cyclic tridiagonal matrix [closed]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...

**2**

votes

**1**answer

78 views

### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...

**1**

vote

**1**answer

45 views

### range of singular values of sub-matrices

Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...

**2**

votes

**0**answers

120 views

### Can the matrix exponential be equal to the elementwise exponential [closed]

Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential $e^A=\sum_{k\...

**1**

vote

**3**answers

164 views

### The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...

**1**

vote

**1**answer

114 views

### Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A &...

**2**

votes

**2**answers

948 views

### Encoding vectors of size $n$ in matrices which less than $2n$ rows [closed]

I have a set of vectors and each has $n$ nonnegative entries.
Moreover, each entry of a vector has a quality: (1) or (2). It makes $2^n$ different possible patterns.
For example, let's take two ...

**1**

vote

**0**answers

42 views

### Explanation for a spectral measure [closed]

Could someone help me, please, to understand in term of entries of a Matrix $M=(m_{i,j})_{i,j\in\{1,n\}^2}$ the following measure :
$$ \frac1{n} \sum_{i=1}^n \langle v_i,e_j \rangle \delta_{\lambda_i}...

**3**

votes

**0**answers

166 views

### Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix
$$A = \left(\begin{array}{}
1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...

**0**

votes

**1**answer

23 views

### Identifying winner of tournament (A,B testings) with none binary test result [closed]

I'm trying to setup a tournament based on votes. Let say user vote for products A, B and C. Each user is presented all possible combinations of products in random order and he picks his preferred ...

**3**

votes

**0**answers

92 views

### How do I ensure that my matrix is positive definite? [closed]

I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form
\begin{equation}
M=\begin{pmatrix}
\Sigma&P'\\
P&\Sigma
\end{pmatrix}
\end{equation}
where $\Sigma$ is a $n\...

**0**

votes

**0**answers

43 views

### How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...

**1**

vote

**1**answer

92 views

### Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists.
Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory).
Preferably the ...