All Questions
73 questions
1
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0
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172
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A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
- 2,712
1
vote
1
answer
608
views
The state-transition-matrix of a physical system,
Here's a simple but potential research problem that I am learning about.
Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
- 1
2
votes
2
answers
123
views
Behavior of matrix rank under thresholding of its elements
Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...
- 2,246
1
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0
answers
947
views
max min of ratio of quadratic forms
Consider the optimization over two vectors $x$ and $y$
$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$
for two positive definite matrices $A$ and $B$.
This problem can be ...
7
votes
2
answers
507
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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, $...
- 103
0
votes
1
answer
246
views
$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?
Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if $\...
- 151
3
votes
2
answers
643
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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}
B_{11}&...
- 752
1
vote
3
answers
5k
views
Number of parameters needed to specify a Hermitian matrix of rank r.
Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank $...
- 199
7
votes
1
answer
250
views
approximate stationary distributions of a doubly stochastic matrix and its supports
Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
- 363
8
votes
1
answer
911
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)$ ...
- 153
2
votes
0
answers
1k
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 ...
1
vote
0
answers
631
views
Bounding the largest Singular value
D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
$(D+B)^{-1}D^2(...
1
vote
0
answers
76
views
When is $F(X)BF(X)$ operator monotone, if $F(X)$ is operator monotone?
Let $\Omega_{n}$ denote the cone of $n\times n$ real symmetric positive definite matrices, and consider $F:\Omega_{n} \mapsto \Omega_{n}$. For $X,Y \in \Omega_{n}$, the matrix valued function $F(\cdot)...
- 1,538
3
votes
0
answers
611
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Can anyone help me deduce a matrix inequality?
The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying $F^{...
- 103
3
votes
0
answers
56
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263
1
vote
0
answers
137
views
Boundary of pseudospectra
Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
- 151
0
votes
1
answer
546
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 ...
- 111
3
votes
0
answers
70
views
Condition number after some "non standard" transform
Given a positive definite matrix $A$, and a diagonal matrix $B$ with positive diagonal entries, is the following inequality generally true?
$$\kappa((A + B)(I + B)^{-1}) \leq \kappa(A)$$
$I$ is an ...
0
votes
3
answers
1k
views
Convex Combination of 2 hermitian matrices
Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...
- 1,421
1
vote
0
answers
113
views
Is my particular finite dimension Toeplitz matrix always strictly positive?
Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...
- 21
1
vote
0
answers
127
views
Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil
Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...
- 757
2
votes
1
answer
304
views
Simultaneous decomposition of three projectors
A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal ...
- 363
1
vote
1
answer
136
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, ...
- 363