All Questions
Tagged with matrices matrix-theory
248 questions
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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
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146
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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
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0
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106
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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
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79
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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-...
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0
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84
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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
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0
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373
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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
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57
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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 $$\text{dim} [\text{span}\{A_1v, A_2v, \...
- 131
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225
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
1
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70
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Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
- 61
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132
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Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
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101
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Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
1
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0
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45
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Matrix Product Chain Representation of an Addition Chain [closed]
I am looking for references to anything interesting thats know about matrix product chains that take the vector $\{1\}$ to another vector $\{n\}$ (the end result of an addition chain). Each matrix ...
- 97
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163
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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
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319
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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 ...
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0
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78
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find linear approximation of non-linear matrix transform [closed]
I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...
- 11
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172
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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
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0
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122
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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
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43
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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
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119
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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
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129
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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
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260
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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
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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
1
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0
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52
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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
1
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0
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137
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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
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113
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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
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305
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how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]
Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
$$...
- 197
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72
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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
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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
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1
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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
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1
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91
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Choosing the best submatrix
Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as
\begin{align}
B_{i,j} =
\begin{cases}
A_{i,j}, & i\in\...
- 287
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1
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224
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Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
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1
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77
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$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$
Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m ...
- 39
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1
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572
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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
0
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1
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170
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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|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...
- 323
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546
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Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
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1
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666
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Kernel of a projection
Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define
\begin{...
- 133
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0
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32
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Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
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131
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On a matrix equation with Kronecker product
Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
- 271
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108
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On the exponentiation of a stochastic matrix where the exponent is a function of matrix size
In this question, I asked about any arbitrary stochastic matrix $A(n)$ of the particular form
$$A(n) = \begin{pmatrix} 1 & 0 & \cdots & 0\\ x_{21} & x_{22} & \cdots & 0\\ \...
- 271
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167
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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
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92
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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
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101
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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
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0
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59
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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
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0
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96
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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
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0
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142
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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
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0
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160
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$l_{\infty}$ 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
-1
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1
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230
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How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]
(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...
- 181
-1
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1
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195
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Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246