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Question 1

4votes

1answer

124views

Most general Logarithm of a matrix

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 ...

lambda

43

askedNov 21 at 11:38

Question 2

5votes

1answer

229views

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

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: ...

pvrnothere

61

askedNov 5 at 10:07

Question 3

0votes

0answers

71views

Ratio of eigenvector amplitudes

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{...

Morgan

1

askedNov 4 at 23:34

Question 4

3votes

1answer

84views

Factorizing a matrix as a product of band matrices

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 ...

Yarden Sheffer

133

askedOct 20 at 11:21

Question 5

0votes

0answers

31views

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

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 ...

Nils R

61

askedOct 16 at 19:16

Question 6

1vote

0answers

12views

Discrepancy in inverse calculated using GHEP and HEP

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 ...

prananna

31

askedOct 5 at 6:04

Question 7

3votes

1answer

282views

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

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 $\...

bvbarsenal

85

askedSep 20 at 22:49

Question 8

0votes

0answers

72views

Motivation of decomposition theorem about linear algebra

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 ...

user1274233

867

askedSep 4 at 5:30

Question 9

5votes

2answers

264views

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

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) ...

gb628782

123

askedAug 21 at 18:52

Question 10

2votes

1answer

89views

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

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{...

Ruben g.

23

askedAug 15 at 15:22

Question 11

0votes

3answers

101views

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

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} - \...

Olumide

1,277

askedAug 13 at 15:50

Question 12

1vote

0answers

55views

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

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 $U$ ...

Raidasam

11

askedJul 25 at 4:44

Question 13

0votes

0answers

22views

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

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?...

Sciencemaster

117

askedJul 15 at 18:05

Question 14

0votes

0answers

44views

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

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....

J0BYJ0

11

askedJul 6 at 4:42

Question 15

1vote

1answer

82views

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

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 $E \in \mathbb{C}^{n \times n}$ ...

J0BYJ0

11

askedJul 1 at 5:21

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Stack Exchange Network

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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