All Questions
Tagged with matrices linear-algebra
1,683 questions
7
votes
2
answers
3k
views
Factorizing a block symmetric matrix
Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to ...
- 20.2k
3
votes
2
answers
2k
views
Eigenvalues of sum of an adjacent matrix and a constant
$A$ is an adjacent matrix of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector.
I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all ...
- 185
28
votes
4
answers
5k
views
Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 4,466
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
- 41
1
vote
1
answer
719
views
A question on gauge functions
In the second paragraph on Page 71 of the book Matrix Analysis by
Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
...
- 13
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
1
vote
2
answers
747
views
Existence of polynomial equation system solution
For $1 \leq i \leq n$, let
$A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn} \\
\end{bmatrix}$, $B_i=\begin{bmatrix} b_{i1} \...
- 57
2
votes
2
answers
819
views
Computing the multiplicity of an eigenvalue of a 0-1 symmetric matrix...
When we want to compute the multiplicity of an eigenvalue of a 0-1 symmetric matrix (viewed as the adjacency matrix of an undirected regular graph), we commonly resort to the know lemma of Feit and ...
8
votes
3
answers
1k
views
Are nilpotent orbits degenerations of semi-simple orbits ?
"Examples first:"
Consider so(3,C). (Co)Adjoint Orbits can be described by equations
x^2+y^2+z^2 = R.
R=0 - is nilpotent cone - algebraic closure of the orbit of nilpotent element. (It is union of ...
- 24.8k
12
votes
3
answers
4k
views
Status of Hadamard matrix conjecture
I would like to know if any progress has been made on Hadamard conjecture :
Hadamard matrix of order $4k$ exists for every positive integer $k$.
- 121
7
votes
1
answer
539
views
A Linear Algebra Problem
Given a matrix $A\in \mathbb{R}^{n\times n}$, I am looking for
a symmetric matrix $S\in\mathbb{R}^{n\times n}$ such that
$$
S A + A^T S = I
$$
$A$ can be assumed to be regular (with positive ...
- 979
2
votes
1
answer
313
views
Derivation of Iteration Rules
Suppose we are given a matrix $V$ and our goal is to find non-negative matrices $W$ and $H$ such that $V \approx WH$. So we want to minimize $K(V || WH)$ (Kullback-Leibler Divergence) where $$K(V||WH) ...
- 21
8
votes
0
answers
694
views
Path connected set of matrices?
Consider the collection of $n$ by $n$ matrices
$$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$
where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint ...
- 1,533
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
- 29
3
votes
1
answer
829
views
polynomial matrices and its spectrum
Hello, all!
I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$,
$$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g _{0,l-1}(...
11
votes
0
answers
305
views
Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial
This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But....
Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
- 23k
5
votes
1
answer
1k
views
Algebra - Decomposition of a matrix polynomial
Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer.
What is known about a possible ...
- 51
2
votes
1
answer
453
views
eigen-decomposition of a special companion matrix
I have a special type of companion matrix, where the "special" part is that each element in the matrix are matrices. For instance, the diagonal with "1":s is instead a diagonal with identity matrices, ...
- 23
2
votes
2
answers
599
views
Eigenvectors of a diagonalizable matrix
Suppose we have a n-by-n symmetric matrix K which can be factorized in a way, K = H * L * H', where L is a m-by-m diagonal matrix and H is a n-by-m matrix. In addition, let's assume n <= m.
Can we ...
- 21
10
votes
1
answer
4k
views
Special considerations when using the Woodbury matrix identity numerically
Are there any special considerations when using the Woodbury matrix identity numerically? What is the best metric for numerical stability in this case? Can anyone point me to a good reference?
The ...
- 211
4
votes
1
answer
1k
views
Solving the matrix equation $XX^t = A$ for binary matrix $X$
How to find all matrices $X \in \{0,1\}^{n \times m}$ that satisfy these equations?
$$X X^t = A \\ \sum_{j=1}^m x_{ij} = 2$$
These articles maybe could help us:
Completely Positive Matrices
Solving ...
- 175
1
vote
3
answers
640
views
Eigenvalues of Krylov matrices
Let an $n\times n$ matrix ${\bf A}$, the all ones vector ${\bf w}$, and the $n\times n$ Krylov matrix
$${\bf K}_n = \left[ {\bf w}\;\;{\bf A}{\bf w}\;\;\ldots \;\; {\bf A}^{n-1}{\bf w}\right].$$
Is ...
- 449
5
votes
1
answer
418
views
positive hermitian elements in $M_n(\mathbb{C})$
Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers
have some special properties:
(i) they are closed under sum,
(ii) they are closed under multiplication by positive ...
- 179
2
votes
1
answer
2k
views
How to do (m)Gram-Schmidt orthogonalization with integers ? (real life problem) ("mathematicalized reformulation")
New edition of the question, "mathematicalized" (thanks to Gerhard).
Consider and integer valued n*n matrix M, with integers elements in the range -N < m < N.
I want to find integer-valued ...
- 24.8k
1
vote
1
answer
2k
views
Find vector in R^n which is orthogonal to given (n-1) vectors v_i under condition v_i are orthonormal.
If we need to find vector in R^n which is orthogonal to given (n-1) vectors,
this is basically solving linear system of equations and can be done in O(n^3) operation.
I wonder is there some ...
- 24.8k
8
votes
1
answer
5k
views
Constructing a unitary matrix
Setting:
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: $Q^{...
- 605
0
votes
1
answer
2k
views
True divide and conquer inversion of large matrices
In https://math.stackexchange.com/questions/2735/solving-very-large-matrices-in-pieces there is a way shown to solve matrix inversion in pieces. Is it possible to turn it into a true divide and ...
Countably Infinite
10
votes
2
answers
7k
views
Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?
$\DeclareMathOperator\Tr{Tr}$Let $A_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a_i<1$ and $\...
- 103
25
votes
5
answers
2k
views
When is a matrix power nonnegative
The following question came up today during a discussion:
Suppose $A$ is an $n \times n$ real matrix. Is there some way to tell whether there exists an integer $q > 0$ such that $A^q$ is ...
- 28.6k
5
votes
1
answer
641
views
Characterizing invertible nonnegative matrices with bounded sums
Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
- 28.6k
2
votes
2
answers
207
views
Families of quadratic Hamiltonians
Hi. What type of 2n dimensional real symmetric matrices can be diagonalized with symplectic transformations (meaning M->SMS^T, S^T means transpose and S is an element of the 2n dimensional real ...
- 23
2
votes
1
answer
351
views
is there a way to solve the following equation?
(I tried asking that on math.stackexchange.com, but did not get a satisfying answer. I am trying here as well, in case someone here will have more insight. The question was eventually abandoned there. ...
- 21
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301
3
votes
1
answer
624
views
Counting matrices with different determinants
Let $A$ and $B$ be two matrices of order $n$ over a finite subset of integers $S$ such that $A$ and $B$ are positive-definite, nonsingular and symmetric.
I am interested in proprieties about $A$, $B$ ...
- 3,463
14
votes
2
answers
2k
views
Finding minimum (or maximum) element of a low rank matrix.
Let $A\in\mathbb{R}^{n\times n}$ and suppose that $A$ is of rank $m\leq n$. Moreover suppose we know $u_1,\ldots, u_m \in\mathbb{R}^{n\times 1}$ and $v_1,\ldots, v_m \in\mathbb{R}^{n\times 1}$ such ...
- 3,217
3
votes
1
answer
523
views
Linear and Isometric Automorphism Groups of the PSD Cone
Let $S_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.
...
user2529
7
votes
1
answer
5k
views
How much can a diagonal matrix change the eigenvalues of a symmetric matrix?
Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that ...
- 449
6
votes
1
answer
448
views
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Dear community,
I have the following combinatorial question which I will explain in short first and then with some more detail. At the end you will find a very simple example.
Short version
Le $A \...
- 199
6
votes
6
answers
11k
views
roots of polynomial with matrix coefficients
Are somewhere existed a method for solving a linear equations over matrices?
For example, I have a task that is similar to next:
find $l \times l$ - matrix $A \in M_{l \times l}(\mathbf{F}_q)$ over $\...
spk
3
votes
1
answer
362
views
Eigenvalues of certain positive matrices
For a matrix $ Q = (q_{ij}) \in GL_n(\mathbb{C}) $ let
$ \overline{Q} = (\overline{q_{ij}}) $ be the matrix obtained by entry-wise complex
conjugation (equivalently, $ \overline{Q} $ is the ...
- 103
16
votes
1
answer
2k
views
Overlapping Gershgorin disks
We all know Gershgorin's Circle Theorem, which I will summarise for convenience. Let $A=(a_{ij})$ be an $n\times n$ complex matrix. Define the disks $D_1,\ldots,D_n$ by $$D_i = \Bigl\{ z : |z-a_{ii}|\...
- 37.7k
9
votes
3
answers
4k
views
2-norm of the upper triangular "all-ones" matrix
Let $M_n$ be the $n\times n$ matrix
$$
(M_n)_{ij}=\begin{cases}1 & i\leq j,\\\\ 0 &i>j.\end{cases}
$$
Is there around an explicit expression or at least an asymptotic for $\left\Vert M_n \...
- 20.2k
12
votes
1
answer
3k
views
Matrix inversion lemma with pseudoinverses
The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.
Suppose we pick $n$ values $x_1,\...
- 28.6k
8
votes
2
answers
1k
views
Solving the equation $xax=b$ in a C*-algebra.
Let $a, b\in A_+$ be positive elements of some C*-algebra $A$.
Assume furthermore that $a$ is invertible.
Is it true that
$$
\exists! x\in A_+\quad:\quad xax=b\quad ?
$$
Already in the case $A=M_2(\...
- 43.2k
5
votes
1
answer
1k
views
Ask some matrix eigenvalue inequalities.
Let $ \begin{bmatrix}
A& B \\\\ B^* &C
\end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$.
Are the following plausible inequalities true? I have seen a lot of ...
- 1,858
10
votes
1
answer
5k
views
Eigendecomposition after multiplying by diagonal matrix
Hello,
If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
- 103
2
votes
2
answers
402
views
Maximization of a matrix product by iterative methods
This might not be very difficult, but I think I may have gotten a little confused.
Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...
- 949
5
votes
1
answer
3k
views
Is there a closed form expression for the inverse of the matrix with elements $A_{i,j}=x_i$ for $i=j$ and $A_{i,j}=1$ for $i\neq j$?
Hello All
Consider a matrix with elements:
$A_{i,j}=x_i$ for $i=j$
$A_{i,j}=1$ for $i\neq j$
Is there a closed form expression for the elements of $A^{-1}$?
Will be glad to know of any reference.
...
- 53
14
votes
3
answers
3k
views
Diagonalizing a Certain Real and Symmetric Toeplitz Matrix
Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by
$$
A_\alpha := \begin{bmatrix}
1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\
\alpha ...
- 3,626
16
votes
6
answers
13k
views
Showing block diagonal structure of matrix by reordering
Suppose we have a block-diagonal matrix $M$, but the block diagonal structure is not immediately apparent from looking at the matrix because the rows/columns are shuffled.
I wish to find a reordering ...
- 911