All Questions
Tagged with linear-algebra matrix-analysis
38 questions
17
votes
2
answers
4k
views
Optimizing the condition number
Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
- 96.4k
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222
6
votes
1
answer
777
views
Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
- 356
11
votes
4
answers
5k
views
The derivative of the Cholesky factor
Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
- 620
5
votes
2
answers
2k
views
How to calculate the square root of matrix $A+B$ perturbatively?
$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...
- 977
3
votes
1
answer
455
views
Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE
$$f(t)=\langle\exp(-...
- 2,252
3
votes
1
answer
381
views
Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices
Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\geq 0$ then $\bigl(\|(a_{ij},b_{ij})\|...
- 175
2
votes
0
answers
92
views
Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices: follow-up
I asked the following question here: "Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\...
- 175
2
votes
0
answers
372
views
What is the Birkhoff norm of a Perron vector?
Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...
- 6,962
1
vote
1
answer
462
views
Is this a full rank matrix? [closed]
According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
- 356
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
13
votes
1
answer
1k
views
A generalization of the Powers-Stormer inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
- 2,019
13
votes
2
answers
946
views
Computing a large permanent
Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
- 6,962
12
votes
0
answers
218
views
Which ordering of factors is needed to obtain this kind of determinantal inequalities?
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
11
votes
2
answers
2k
views
Inverse of a small submatrix
Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
- 241
11
votes
2
answers
3k
views
Do singular values dominate eigenvalues?
Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots \...
- 1,135
11
votes
1
answer
1k
views
Show that these vectors are linearly independent almost surely
So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question.
Problem: I have $m<n$ real $...
- 344
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 716
10
votes
2
answers
5k
views
Nuclear norm as minimum of Frobenius norm product
Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix.
It is claimed that
$$
\|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
- 2,239
10
votes
1
answer
630
views
Minimum distance of a symmetric matrix to diagonal matrices
Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix and $n\geq 2$. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for ...
- 4,484
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.6k
6
votes
1
answer
737
views
Rank of the absolute-value matrix $|M|$ vs. rank of $M$
Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).
Let $|M|$ be the matrix obtained by taking the absolute value of each entry ...
- 20.2k
6
votes
2
answers
880
views
Matrix-convexity of inverse of the cofactor matrix
Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...
- 3,383
5
votes
3
answers
693
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$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 A\|\le\frac{\...
- 153
5
votes
0
answers
586
views
A minimal eigenvalue inequality
Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
- 2,239
5
votes
3
answers
3k
views
Spectral properties of the LDL^T matrix factorization
Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...
- 757
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 \...
- 41
4
votes
1
answer
720
views
Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
- 287
4
votes
2
answers
311
views
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$
For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix.
I would like to solve the following equation for the ...
- 1,538
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 145
2
votes
1
answer
242
views
Necessary conditions for existence of linear combination of these matrices to be singular
I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated.
QUESTION
Let $P_1,\dots,P_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ ...
- 344
1
vote
1
answer
1k
views
product of Gaussian random matrix and a deterministic diagonal matrix
Suppose that $G$ is an $n\times n$ Gaussian random matrix of i.i.d. entries $N(0,1/n)$ and $D$ is an $n\times n$ deterministic diagonal elements. I'd like to know if there have been results on the ...
- 640
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} \...
0
votes
1
answer
533
views
Follow up: Show that these vectors are linearly independent almost surely
I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I want to discuss regarding it. Unfortunately I can'...
- 344
-2
votes
1
answer
1k
views
Derivative of log determinant [closed]
Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right).
$$
- 245