Questions tagged [matrix-theory]
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
466 questions
3
votes
1
answer
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Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
- 355
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
5
votes
1
answer
2k
views
Diagonalization of real symmetric matrices with symplectic matrices
Consider the following real symmetric matrix
$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$
Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
- 51
1
vote
0
answers
260
views
Product of exponentials of matrices
Let $A$ and $B$ be commuting $n\times n$ positive definite complex matrices and let $C$,$D$ be other complex matrices. I wish to think of $C,D$ as small and $A,B$ as any size.
Suppose we are looking ...
- 280
4
votes
0
answers
112
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Inducing surjections on $GL_n(-)$?
Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
- 61
1
vote
1
answer
101
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Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix?
Suppose $R$ is a (possibly noncommutative) ring. I was thinking of $R=S[x_1,\ldots,x_n]$ or $R=S[x_1,x_1^{-1},\ldots,x_n,x_n^{-1}]$ for $S$ some (possibly noncommutative) ring. Now, let $GL_2(R)$ be ...
- 348
5
votes
2
answers
2k
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Decomposing a matrix into a product of sparse matrices
How to study the decomposition of a square matrix into a product of sparse matrices?
There are no restrictions on the number of matrices in the product, but the fewer the better.
- 451
-1
votes
1
answer
61
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Finding a matrix with shared eigen vectors with a given matrix [closed]
If I have a known matrix A, is there a method to find a matrix B that share all the eigen vectors of Matrix A?
- 9
3
votes
1
answer
3k
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Solving a vector of quadratic equations
I have a system of $n \times 1$ equations
$$
0 = A\,vec(xx^t) + B x + C
$$
where
$x$ is a $n \times 1$ vector of unknowns
$x^t$ means transpose
$vec$ means $xx^t$ has been vectorized so has dimension ...
- 43
3
votes
2
answers
132
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Vanishing zeroes in matrix powers
For a matrix $M\in\mathbb{R}^{n\times n}_{\geq 0}$ with nonnegative entries, we define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive (if there is one).
...
- 4,829
1
vote
0
answers
127
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Low-rank approximation of sub-sampled matrix
Considering a large data matrix $X$ with zero-centered columns that is assumed to be approximately low-rank, it is common to do PCA and project the data onto the top few principal components, and use ...
- 111
10
votes
2
answers
1k
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When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?
Suppose $A_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$).
If $\sum_{k=1}^\infty A_k$ exists, then
$\sum_{k=1}^\infty \|A_k\| < +\infty$,
Where $\|A\|=\...
- 173
2
votes
1
answer
303
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Submatrix with small sum of elements
Let $A$ be an $n \times n$ matrix, for which I know the size of the sum of all its entries. Now I want to select an $m \times m$-submatrix, whose sum of entries is as small as possible. Is there any ...
- 2,207
4
votes
0
answers
437
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A sum of Ramanujan sums
I have the following question about Ramanujan sums.
(All vectors and matrices here will be understood to have integer entries.)
Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...
- 1,381
5
votes
4
answers
2k
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Differentiability of eigenvalue and eigenvector on the non-simple case
Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
- 1,638
3
votes
0
answers
248
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Exponential of tridiagonal matrix and boundedness of successive quotients of components
Let $a\geq 0,b>0$, $N\in \mathbb N$ and
$$ A_N = \begin{pmatrix}a(-N) & b && 0\\b & \ddots & \ddots & \\ &\ddots & \ddots & b\\0 && b & a N \end{...
- 85
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votes
2
answers
392
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Which unimodular matrices are integer combinations of permutations?
Consider the set of all integer linear combinations of permutation matrices of some fixed dimension. Is there a description of the set of unimodular matrices in this lattice?
- 481
3
votes
2
answers
570
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Matrices Representing Bounded Operators and Absolute Values
Let $A=(a_{ij})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. For every positive integer $n$, we shall denote with $A_n$ the $n \times n$ matrix $A_n=(a_{i,j})_{i,j=1}^{n}$, and if $x \...
- 640
2
votes
0
answers
412
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Bound on eigenvalues of A+B (Hermitan matrices) which is better that the Lidskii and Weyl bounds
I have two positive definite $N\times N$ Hermitian matrices $A$ and $A$ and am interested in bounding the eigenvalues of $A+B$ in terms of the eigenvalues of $A$ and $B$. Let $\lambda_k(\cdot)$ be ...
3
votes
0
answers
70
views
Condition number after some "non standard" transform
Given a positive definite matrix $A$, and a diagonal matrix $B$ with positive diagonal entries, is the following inequality generally true?
$$\kappa((A + B)(I + B)^{-1}) \leq \kappa(A)$$
$I$ is an ...
6
votes
1
answer
586
views
On a trace condition for positive definite $2\times 2$ block matrices
Consider the following block matrix
$$
X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix},
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}...
- 2,712
3
votes
1
answer
240
views
Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
- 1,813
4
votes
1
answer
545
views
No arbitrary product of matrices has eigenvalue 1?
Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d_{i,i}=0,\forall i$.
The matrices $A_i$ are constructed from Identity matrices, $I_{n*n}$...
- 143
2
votes
2
answers
307
views
The structure of the $n$-th power of a special matrix
Assume the following matrix
$$
C_p^{(a,b)}:=\left(
\begin{array}{cccccc}
a &a &0 &\cdots &\cdots &0 \\
0 &0 &a &\ddots &\ddots &\vdots \\
\vdots &\ddots &...
- 313
0
votes
1
answer
460
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A question on orthogonal matrix [closed]
Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.
- 377
0
votes
1
answer
572
views
Recurrence Equation and Matrix Convergence
To begin with, let us give the conceptual background needed to expose the problem. First of all, we shall consider the set $\mathbb{L}^{n} = \mathbb{R}^{n}_{\geq0} = \{\overrightarrow{x}\in\mathbb{R}^{...
user89024
1
vote
1
answer
167
views
Find a matrix and its inverse satisfying lower and upper bounds
I reduced a problem of matrix completion to the problem
find $A,B$ such that
$AB=I$
$A_{min}\leq A \leq A_{max}$
$B_{min}\leq B \leq B_{max}$
One possible approach would be to just minimize $\|AB-...
- 11
0
votes
1
answer
170
views
Non-strict column diagonally dominant matrix inner product
Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...
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3
votes
1
answer
565
views
Second-order perturbation expansion for singular value decomposition
Let $A = U\Sigma V^T$ be the singular value decomposition (SVD) of a $n\times m$ matrix $A$. Let $\tilde{A} = A + \epsilon P$ be a perburbation of $A$. It is possible, using tools from Matrix ...
- 995
3
votes
0
answers
65
views
Why does the objectivity rule out the convexity?
In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...
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2
votes
1
answer
122
views
Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries
Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...
- 1,429
1
vote
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 ...
1
vote
0
answers
52
views
Matrix transformation [closed]
I want to show that
$(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis
with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$
is equivalent to $M_\gamma$ has no purely imaginary ...
- 11
2
votes
2
answers
534
views
Number of Skew Symmetric Matrices of fixed rank
The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g.
http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf
Is the number of skew-...
- 21
6
votes
2
answers
825
views
Explicit solution to a Rayleigh quotient equation
For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
\biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...
- 163
1
vote
1
answer
807
views
Restricted Isometry Property for Discrete Fourier Transform Matrix
I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...
- 1,127
0
votes
0
answers
52
views
Gramian of a permutation group orbit
let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure:
$$
W^TW=(P_{1}t,\cdots,P_{k}t)
$$
with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...
- 329
2
votes
4
answers
293
views
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle
...or prove that none exists.
Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory).
Preferably the ...
- 21
1
vote
1
answer
607
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
1
vote
0
answers
119
views
inverse of asymptotic Toeplitz matrix with band limited associated function
I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
...
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0
votes
1
answer
246
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$P(Z)$ 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
5
votes
1
answer
725
views
Optimization of a function of a positive definite matrix and its inverse
This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.
Suppose I have two real, positive ...
- 51
3
votes
2
answers
2k
views
A question on determinant of a matrix polynomial
Let
${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.
${\rm{P(}}\lambda {\rm{) = }...
- 151
1
vote
1
answer
352
views
Fixed point of quantum operations
A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
- 35
9
votes
3
answers
390
views
Is there a standard name for the following type of linear operator?
Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
- 38.6k
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
1
vote
0
answers
121
views
Limits on a parameter $\alpha$ to get positive definite matrix
Given positive semi-definite $n\times n$ matrices $B_k$, how would I go about getting the limits on $\alpha_k$ such that the expression
\begin{equation}
\mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k
\end{...
- 35
0
votes
1
answer
285
views
Matrix inequality between a traceless matrix and identity
Given a traceless matrix $C\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$.
- 35
1
vote
1
answer
940
views
Uniqueness and invariance of the LDLT decomposition
A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
- 101
8
votes
2
answers
12k
views
Relation between eigenvalues of $A$ and $A^TA$?
For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...
- 181