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
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Eigenvector localizaiton
I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry ...
- 6,962
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52
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Efficiently Updating Matrix Multiplication Result When One Matrix Changes [closed]
Suppose you have two matrices $A \in Z_q^{m\times l}$ and $B \in Z_q^{l\times n}$, and the product $A\cdot B$ has already been computed. Now, matrix $B$ remains unchanged, but a few elements in matrix ...
- 11
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45
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Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations
Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
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Correlation Matrix Problem of Three Decomposition Level of DWT
I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix.
The step I'm trying to follow is:
The DWT coefficientes are obtained from ...
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Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
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When can we separate two pairs in ${\mathbb H}_n$, although it is not a lattice?
Recall that a lattice is a partially ordered set $E$ for which any pair $a,b\in E$ admits a least upper bound and a greatest lower bound. Remark that given four elements $a_i,b_j$ ($j=1,2$), in order ...
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Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures
TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...
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Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
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Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
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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 ...
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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 & \...
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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 ...
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Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?
For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes
$x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
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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 ...
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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
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Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
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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 ...
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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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Does there exist a canonical form for normal matrices which extends the following embedding?
Given an unordered pair of complex numbers $\{w,z\}$, we can associate to it the complex matrix
$$\frac 1 2\left[\begin{matrix}w + z + \frac{\left(w - z\right)^{2} + \left|{w - z}\right|^{2}}{2 \left|{...
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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 ...
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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 ...
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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 $$\text{dim} [\text{span}\{A_1v, A_2v, \...
- 131
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Spectral theorem for symmetric real tensors
Is there a definition of eigenvalues that allows to use a spectral theorem?
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
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Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero
I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $\...
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A lower-bound on matrix-function with vector product
I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
- 5,405
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Upper and lower bounds on the entries of a matrix power
Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds:
$$A_{ij} > 0 \iff A_{ji} > 0.$$
I know that no more ...
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Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
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Convergence rate of Toda/Morse flow
Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow
\begin{align}
\frac{dA}{dt} &= \left [ C\circ A , A \right ] \\
A(0) &= A_0 \ .
\end{align}
...
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On norm convergence of a commutator and implications
Given a sequence of real symmetric $(2^N\times 2^N)$- matrices $(H_N)_{N\in\mathbb{N}}$ on the $N$-fold tensor product of $\mathbb{C}^2$ with itself, such that
\begin{align}
\lim_{N\to\infty}||[H_N,...
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Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
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Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
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What is the expected inverse of 1 plus a Wishart distribution?
Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$
Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...
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56
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Perturbations of the eigen/singular directions
Let $A = U_A \Sigma_A V_A^\top$ and $B = U_B \Sigma_B V_B^\top$, and $A+B = U \Sigma V^\top$ be the respective singular vector decompositions.
Is there some known relationship of the form
$$\| U_A ...
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Condition for the integrability of a matrix function
Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable?
$$
I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\...
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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 ...
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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)...
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Matrix Bernstein for spherical random variables
Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
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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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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 ...
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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
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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 ...
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Maximum singular value of sum of an Hermitian and an anti-Hermitian matrix
Let $H$ be an $n\times n$ Hermitian matrix and $A$ an $n\times n$ anti-Hermitian matrix, i.e. $H^\dagger = H$, $A^\dagger = -A$. Consider their sum $S= H+A$. Let $\{\sigma_i(S)\}_{i=1,\dots,r}$ denote ...
- 11
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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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76
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When is $F(X)BF(X)$ operator monotone, if $F(X)$ is operator monotone?
Let $\Omega_{n}$ denote the cone of $n\times n$ real symmetric positive definite matrices, and consider $F:\Omega_{n} \mapsto \Omega_{n}$. For $X,Y \in \Omega_{n}$, the matrix valued function $F(\cdot)...
- 1,538
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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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0
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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