# Tagged Questions

**11**

votes

**1**answer

172 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**1**

vote

**1**answer

69 views

### Connection between eigenvalues of A and its LDL decomposition

Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n_{ij}$: $n_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n_{ij} = 0$ otherwise. Consider $A_{ij} \equiv ...

**4**

votes

**1**answer

49 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, ...

**7**

votes

**1**answer

209 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 & ...

**4**

votes

**2**answers

200 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

**1**answer

62 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 ...

**0**

votes

**1**answer

49 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

123 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 ...

**0**

votes

**0**answers

93 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

**3**answers

146 views

### Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ ...

**0**

votes

**1**answer

76 views

### Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...

**0**

votes

**0**answers

89 views

### Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition.
What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...

**19**

votes

**0**answers

624 views

### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...

**3**

votes

**0**answers

131 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!

**-1**

votes

**1**answer

141 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.,
...

**4**

votes

**1**answer

112 views

### variation of the Lieb concavity theorem

A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:
$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$
for $s \in ...

**0**

votes

**1**answer

81 views

### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...

**5**

votes

**0**answers

257 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

154 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 ...

**5**

votes

**1**answer

135 views

### Simultaneous Tridiagonalization of a given set of hermitian matrices?

I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?

**5**

votes

**2**answers

330 views

### How to check whether a matrix is completely positive or not?

The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...

**2**

votes

**1**answer

84 views

### Known Results on Convexity of Numerical Range

Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set
\begin{align}
\mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\}
\end{align}
...

**7**

votes

**1**answer

162 views

### How much can I perturb a symmetric stochastic matrix and keep positive solutions?

Suppose $A$ is a symmetric stochastic $n \times n$ matrix with least eigenvalue $\lambda_n>0$. Consider real symmetric perturbations $\widehat{A}$.
How large can I take $\epsilon$ such that ...

**4**

votes

**2**answers

220 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**1**

vote

**2**answers

121 views

### Bound on smallest entry of inverse matrix

For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse ...

**5**

votes

**1**answer

127 views

### Are there any known results on numerical ranges of rank-one positive semi-definite matrices?

In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...

**3**

votes

**1**answer

192 views

### When is this matrix singular?

Consider matrix $A$ with $(j,k)$â€²th 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

167 views

### Incoherence of the row/column span

Due to V.Chandrasekaran., et alâ€Ž (p.11) : In general for any $k$-dimensional subspace of $A_{nÃ—n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...

**0**

votes

**0**answers

238 views

### Proving that a certain matrix inverse is always positive definite

Take any positive definite Hermitian matrix $X$.
Put all values of $X$ equal to 0 except for the values along the center $2K+1$ diagonals which are kept untouched. Denote the new matrix by $Y$.
Let ...

**0**

votes

**1**answer

179 views

### Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ...

**3**

votes

**3**answers

386 views

### The derivative of the Cholesky Factor

Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be it's Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...

**3**

votes

**1**answer

97 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**7**

votes

**1**answer

777 views

### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

**2**

votes

**0**answers

106 views

### A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...

**5**

votes

**3**answers

380 views

### Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R ...

**2**

votes

**1**answer

96 views

### Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...

**0**

votes

**1**answer

197 views

### Bounding the positive semi-definite matrix with its block diagonal matrix [closed]

Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{*{20}{c}}
{{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\
...

**3**

votes

**2**answers

255 views

### On a determinant inequality of positive definite matrices

Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following:
$$
B=\left[\begin{array}{ll}
...

**1**

vote

**0**answers

78 views

### M-matrix with nonconstant entries properties

I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 ...

**1**

vote

**0**answers

108 views

### Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...

**1**

vote

**0**answers

85 views

### Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...

**8**

votes

**1**answer

226 views

### Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...

**0**

votes

**0**answers

313 views

### Diagonal of the inverse of a 6x6 symmetric partitioned matrix

Let
$$M = \begin{bmatrix}
A & B \\
B & C
\end{bmatrix}$$
in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...

**4**

votes

**2**answers

159 views

### Maximal norm-1 projection

Suppose I have a real unitary matrix $U$ and a unit vector $\mathbf{x}, \|\mathbf{x}\|_2 = 1$. What is the solution to the following problem?
$$
\widehat{\mathbf{x}} = \arg\max_{\mathbf{x}, ...

**0**

votes

**2**answers

241 views

### Eigenvalues of an amplification matrix

Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something about the eigenvalues ...

**9**

votes

**1**answer

411 views

### M-matrix plus S-matrix is P-matrix?

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...

**11**

votes

**2**answers

394 views

### Computing a large permanent

Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...

**1**

vote

**0**answers

80 views

### Matrices with a common fischer basis

Let $A$ be a real symmetric $n\times n$ matrix normalised such that $Tr[A]=1$. Define 'fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for this ...

**4**

votes

**1**answer

226 views

### Rank of a matrix with missing entries

Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M_{A, B} = 1
$
if $A \subseteq ...

**17**

votes

**1**answer

660 views

### 2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...