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.
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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? ...
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How do I test two square matrices are transpose to each other if only the column vector summations are known?
Given two secret square matrices, say $\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\
{{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\...
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Classifications of the indefinite generalized Cartan matrix
I want to know that the present results about classifications of generalized indefinite Cartan matrices.
I only have known that the classifications of hyperbolic matrces.
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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 ...
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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 $n$.
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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}(...
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The right expansion of a square root matrix
I have some issues in finding an asymptotic expansion for a square root matrix and I have already posted a question (Asymptotic expansion square root matrix). Somebody redirected me to a post where ...
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How to understand the matrix behind a Hamiltonian?
thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...
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Moore-Penrose question
Let $A=BD^{\dagger}B^{T}$. I am looking for conditions under which $A^{\dagger}$ is a "nice" expression in $B$ and $D$ and their Moore-Penrose pseudo-inverses.
Do you know of such conditions?
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A simple matrix multiplication query [closed]
The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
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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 ...
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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 \...
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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?
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A problem that involves matrix and Lorentz Transformation [closed]
To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G A^T=A^...
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adjacency matrix of random geometric graphs [closed]
Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
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How to find matrix representations of a boolean algebra? [closed]
Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...
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