Questions tagged [matrix-decomposition]

Question 1

Consider any matrix $A \in \text{GL}_d(\mathbb{C})$, i.e, a square invertible matrix. We define a logarithm of $A$ as any matrix $X$ such that $$e^X = A.$$ Our objective is to find of possible ...

Question 2

Out of curiosity, I came across the study of matrices of the form $U = B^\top B^{-1}$ for $B \in GL(n,\mathbb{F}_2)$. In essence, U represents how matrice $B$ is asymetric. This raised the question: ...

Question 3

I am studying a generalized eigenvalue problem which can be partitioned as $$ \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{21}}}&{{A_{22}}} \end{array}} \right]\left[ {\begin{...

Question 4

Let $A$ be $n\times n$ matrix. Can $A$ be written as $A=B_1\cdots B_k$ where $B_i$ are band matrices with constant bandwidth, and $k=O(n)$? Is it always possible? Is there an efficient algorithm for ...

Question 5

I have a question about an equation that is so simple that I feel like it should have a name and be analyzed, but I can't find a reference for it, so I am hoping someone here has seen this before. I ...

Question 6

Say we have a matrix $A = L + \beta^{2} M$, where $\beta$ is a real scalar. The matrices $L$ and $M$ are symmetric positive semi-definite and symmetric positive definite respectively. I am interested ...

Question 7

I'm working on a matrix factorization problem and would appreciate insights on the following conjecture: $\forall y,z \in \mathbb{N}$ such that $y > z$, there does not exist column-stochastic $\...

Question 8

When eigenvectors form a basis, they're a very good basis because they transform a given matrix into a diagonal matrix. The problem is that eigenvectors don't generally form a basis. However, if we ...

Question 9

In Golub & Van Loan's Matrix Computations, I came across the following problem and I am stumped (been at it for a few days now). A matrix $M\in \mathbb{R}^{n\times n}$ (not necessarily symmetric) ...

Question 10

Given that $A = LU$, where $L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and $U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{...

Question 11

I understand the LU factorization algorithm as the result of the recurrence relationships $$ u_{ij} = a_{ij} - \sum_{k = 1}^{i - 1} l_{ik} u_{kj} $$ for $i \le j$, and $$ l_{ij} = \left[ a_{ij} - \...

Question 12

I would like to know if anyone knows any reference about rank one update of Schur decomposition. Assume that we know the Schur decomposition of $A,$ which is $A=QUQ^{-1},$ where $Q$ is unitary and $U$ ...

Question 13

If you have a collection of n (nonzero and unique) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them in such a way that 'rules out' alternative forms? ...

Question 14

I am new to lie theory and representation theory. I heard about this interesting factorization known as the Bipolar decomposition which uses the Mostow decomposition. The article is https://www....

Question 15

As the question states: Given $M \in \mathbb{C}^{n \times n}$ where $\det(M) = 0$. Does there exist a real matrix $R \in \mathbb{R}^{n \times n}$ and invertible matrix $E \in \mathbb{C}^{n \times n}$ ...

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Questions tagged [matrix-decomposition]

Most general Logarithm of a matrix

Sufficient conditions for representing the matrice $U$ as $B^\top B^{-1}$

Ratio of eigenvector amplitudes

Factorizing a matrix as a product of band matrices

Interpretation of an equation arising in matrix perturbation on the inner product of eigenvectors, weighted by eigengaps

Discrepancy in inverse calculated using GHEP and HEP

Existence of a kernel-preserving stochastic factorization of a column-stochastic matrix

Motivation of decomposition theorem about linear algebra

If the matrix $A$ is positive definite and $T$ is its symmetric part, show that $x^T A^{-1} x \leq x^T T^{-1}x$

Efficient way to calculate basis of $\text{Null}(A^T)$ from given $LU$ factorisation without calculating $A$.

Struggling to infer the pseudocode of the LU factorization from the algorithm.

Efficient way for finding rank-one update of Schur decomposition?

Does diagonalization of a matrix imply that the eigenvectors are unique to a given form of a matrix? [duplicate]

Geometric interpretation of the matrix exponential of imaginary symmetric and imaginary skew symmetric matrices?

Can a singular complex square matrix be factorized into a complex invertible matrix and a real square singular matrix?

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