All Questions
Tagged with matrices matrix-theory
86 questions with no upvoted or accepted answers
1
vote
0
answers
162
views
Convex matrix combination
We all know the notion of a convex combination like
$$\lambda x_1 \; + \; (1 - \lambda) x_2$$ for some $\lambda \in (0, 1)$.
However, I am trying to find literature where this concept has been ...
- 61
1
vote
0
answers
111
views
Solving a block tridiagonal system with diagonal perturbations
Say we have a block tridiagonal matrix, $T \in \mathbb{R}^{NL \times NL}$, with constant off diagonals, $\mathbf{B} \in \mathbb{R}^{L\times L}$, given by
$$
T = \begin{bmatrix} \mathbf{A}_1 & \...
- 11
1
vote
0
answers
216
views
Schatten norm inequality
Let $A,B$ be two $n\times n$ matrices.
Find a lower bound of the $p$-th Schatten norm
$\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
- 97
1
vote
0
answers
42
views
What is the complexity of the matrix multiplication closure for a given generating system?
Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
- 21
1
vote
0
answers
146
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 573
1
vote
0
answers
106
views
Number of non-singular matrices with entries in $\{1, -1\}$
What is the number of non-singular $n \times n$ matrices with entries in $\{1, -1\}$? Here I assume that two such matrices are equivalent if one can be obtained from the other by permutations of rows ...
- 745
1
vote
0
answers
79
views
There is an observation on the eigenvalues of the sum of a kind of special Hermitian matrices. How to prove it?
Suppose $A$ and $B$ belong to a kind of special hermitian matrices, which have the following properties:
$A$ and $B$ contain only one negative eigenvalue.
the negative eigenvalue and the second-...
1
vote
0
answers
84
views
In matrix product, differentiate one element with respect to another element
Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a ...
- 11
1
vote
0
answers
373
views
Upper bound on the sum of the smallest non-zero eigenvalues
Let $\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$ be a set of symmetric and positive semidefinite matrices.
For a matrix $A_k \in \mathcal A$, denote its (real) eigenvalues by ...
- 245
1
vote
0
answers
57
views
For which sets of linearly independent complex matrices does there exist a vector whose image set under the given matrices is linearly independent?
For which sets $\{A_i \}_{i=1}^n \subseteq M_d$ of linearly independent $d\times d$ complex matrices does there exist a vector $v\in \mathbb{C}^d$ such that $$\text{dim} [\text{span}\{A_1v, A_2v, \...
- 131
1
vote
0
answers
225
views
Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
1
vote
0
answers
70
views
Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
- 61
1
vote
0
answers
132
views
Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
1
vote
0
answers
101
views
Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
1
vote
0
answers
163
views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
- 11
1
vote
0
answers
319
views
Derivative of complex matrix pseudo inverse with respect to real and imaginary components
I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...
1
vote
0
answers
172
views
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
0
answers
122
views
Fullrankness of sum of time shifts
I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
- 93
1
vote
0
answers
43
views
relationships between $AA^T$ and $[(I-A)(I-A)^T]^{-1}$ with $A$ being strictly lower triangular
I have a matrix $A$ which is strictly lower triangular. Now, I am trying to find some general statements/relationships of following matrices $U,D,V,K$ defined as:
$AA^T=VKV^H$,
$[(I-A)(I-A)^T]^{-1}=...
- 21
1
vote
0
answers
119
views
An analogue of Hermitian matrix - does it exist?
Let $k$ be any field and $R\subseteq M_s(k)$ be a subring of $s\times s$ matrices over $k$. Identify $k$ with the scalar matrices, so that $k\subseteq R$. Let $A\in M_n(R)$ be an $n\times n$ matrix.
...
- 3,041
1
vote
0
answers
129
views
Matrix majorization when a diagonal matrix is multiplied from right and left
Let $D_1$ and $D_2$ be two diagonal matrices such that $D_1^2+D_2^2=I$ (identity matrix). Suppose matrix $A$ majorizes matrix $B$. Can we show that matrix $A$ majorizes matrix $D_1 A D_1 + D_2 B D_2$? ...
- 181
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
1
vote
0
answers
127
views
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
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
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
72
views
sharper interlacing
The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I ...
- 6,962
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
0
votes
0
answers
131
views
On a matrix equation with Kronecker product
Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
- 271
0
votes
0
answers
108
views
On the exponentiation of a stochastic matrix where the exponent is a function of matrix size
In this question, I asked about any arbitrary stochastic matrix $A(n)$ of the particular form
$$A(n) = \begin{pmatrix} 1 & 0 & \cdots & 0\\ x_{21} & x_{22} & \cdots & 0\\ \...
- 271
0
votes
0
answers
167
views
Multiplying by Loewner-ordered matrices
Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
- 93
0
votes
0
answers
92
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
0
votes
0
answers
101
views
Find occurrences of certain matrix inside a matrix
This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching.
We know for ...
- 101
0
votes
0
answers
59
views
A system of inequalities involving a skew-symmetric integer matrix
Which skew-symmetric integer matrices $S$ satisfy the following inequalities
$SV_i \ne z_iE_i$ for all $i = 1,\cdots, n$
where
$V_i$ denotes the column with integer entries such that the $i$-th ...
- 356
0
votes
0
answers
96
views
Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality
Let $A$ be an $n \times n$ matrix. Let $A_k$ be the matrix obtained by keeping the first $k$ rows of $A$ fixed and substituting $0$ for the rows $k+1$ to $n$. To be precise, we write $A= [R_1...R_k, ...
- 1,465
0
votes
0
answers
142
views
Jordan Decomposition of Sparse matrix
Suppose we are given $n \times n$ rational matrix, $A$ with at most $k$ nonzero elements in each row and each column with $k \ll n$.
What is the best algorithm to compute its Jordan decomposition? ...
- 1,503
0
votes
0
answers
160
views
$l_{\infty}$ norms of matrix perturbations
Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.
What needs to be the bounds on (which?) norm of $B$ to ensure that $\lambda_{max}(...
- 1,893