All Questions
Tagged with linear-algebra matrix-analysis
364 questions
24
votes
0
answers
1k
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 ...
- 483
2
votes
1
answer
1k
views
Trace inequality for matrices with determinant 1
Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that
$\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$
I suspect that this can be ...
- 13.4k
4
votes
2
answers
292
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 ...
- 42.4k
0
votes
1
answer
1k
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 ...
- 720
0
votes
1
answer
836
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} \...
- 24.2k
2
votes
1
answer
346
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 ...
- 15.8k
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 ...
- 720
6
votes
1
answer
417
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?
- 21.5k
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 ...
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 1,748
3
votes
2
answers
643
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:
$$
B=\left[\begin{array}{ll}
B_{11}&...
- 111
3
votes
0
answers
193
views
Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
- 133
4
votes
4
answers
3k
views
The multiplicity of the max eigenvalue in matrix multiplication
Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
- 133
4
votes
1
answer
299
views
variation of the Lieb concavity theorem
A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B:
$$
(A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \}
$$
for $s \in ...
- 28.6k
6
votes
0
answers
489
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 ...
CommunityBot
- 1
2
votes
0
answers
477
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 ...
- 73
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, ...
- 2,286
3
votes
1
answer
414
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}
...
- 1,421
5
votes
2
answers
2k
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)$
...
- 145
7
votes
1
answer
443
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 $\...
- 1,081
1
vote
2
answers
2k
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 ...
- 6,962
5
votes
1
answer
286
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 ...
- 6,351
3
votes
1
answer
264
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 $\...
- 2,239
2
votes
0
answers
1k
views
Incoherence of the row/column span
Due to V.Chandrasekaran., et al (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...
- 21
0
votes
1
answer
268
views
Nonnegative Matrix
Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\...
CommunityBot
- 1
8
votes
1
answer
911
views
A Problem on Linear Algebra
I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
- 153
7
votes
3
answers
2k
views
Optimization problem on trace of rotated positive definite matrices
Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...
- 7,514
2
votes
0
answers
130
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, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
- 68.9k
2
votes
1
answer
359
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 ...
CommunityBot
- 1
1
vote
1
answer
1k
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}}}\\
...
- 28.6k
1
vote
0
answers
269
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 ...
- 11
5
votes
2
answers
429
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
- 342
2
votes
0
answers
372
views
What is the Birkhoff norm of a Perron vector?
Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...
CommunityBot
- 1
3
votes
0
answers
706
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 ...
CommunityBot
- 1
8
votes
1
answer
603
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, $C$ ...
- 28.6k
0
votes
0
answers
957
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 ...
- 12.1k
5
votes
2
answers
706
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?
$$
\widehat{\mathbf{x}} = \arg\max_{\mathbf{x}, ~\|\...
- 2,545
0
votes
2
answers
737
views
Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something about the eigenvalues ...
- 148k
9
votes
1
answer
1k
views
M-matrix plus S-matrix is P-matrix?
I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...
CommunityBot
- 1
13
votes
2
answers
946
views
Computing a large permanent
Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
- 82.7k
4
votes
1
answer
543
views
Rank of a matrix with missing entries
Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M_{A, B} = 1
$
if $A \subseteq B$;...
- 30.1k
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
1
vote
2
answers
508
views
Sufficient conditions for inverse-positivity
I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...
CommunityBot
- 1
1
vote
0
answers
358
views
Kernel of modified Kronecker sum
The Kronecker sum of two matrices $A \in M(n \times n, \mathbb{R})$ and $B \in M(m \times m,\mathbb{R})$ is defined by the matrix
$$A \oplus B = A \otimes I_m + I_n \otimes B \in M(nm \times nm, \...
- 397
3
votes
2
answers
186
views
Triangularizing a function matrix with smooth eigenvlaues
Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given ...
CommunityBot
- 1
0
votes
1
answer
1k
views
Simultaneous Jordanization
Hello everyone
I would like to have a detailed reference to the statement bellow:
Let $A,B\in \mathbb{R}^{n\times n}$ such that $AB=BA$. Suppose $A$ has real eigenvalues only and $B$ is ...
- 91.8k
0
votes
2
answers
818
views
a question about the Jordan form [closed]
Some reference say that if rank($A$)=rank($A^2$),then the geometric and algebraic multiplicities of the eigenvalues $\lambda=0$ are equal;that is,all the Jordan blocks correspondint to $\lambda=0$ (if ...
- 60.5k
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 $...
- 91.8k
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 ...
- 25.3k
5
votes
2
answers
495
views
Hadamard product and inertia
One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...
- 41