For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...
0
votes
0answers
5 views
Maximum related to Nonnegative Matrix
Suppose $A$ is a fixed nonnegative $n\times n$ matrix (i.e. $A_{ij}\geq0$ for all $i,j$). Then for any arbitrary $n$ positive numbers $x_1,\ldots,x_n$, we denote:
...
1
vote
4answers
36 views
Matrices: left inverse is also right inverse? [duplicate]
If $A$ and $B$ are square matrices, and $AB=I$, then I think it is also true that $BA=I$. In fact, this Wikipedia page says that this "follows from the theory of matrices". I assume there's a nice ...
2
votes
0answers
17 views
Intuiting Product of Elimination Matrices (and NOT by Matrix Multiplication)
I want to intuit, and NOT compute with matrix multiplication, $M:=\color{green}{E_{P_3 \rightarrow P_4}}\color{#CA790F}{E_{P_2 \rightarrow P_3}}E_{P_1 \rightarrow P_2},$ where:
$E_{P_1 \rightarrow ...
3
votes
0answers
43 views
Derivative of trace of a matrix
$$\dfrac{\partial\operatorname{Trace}\left[\left(AB\right)^{T}Q\left(AB\right)\right]}{\partial A}=\text{ ?}$$
where $Q = Q^T>0$
2
votes
0answers
33 views
Determining the Jordan Canonical Form $18\times 18$ matrix
Let A be an $18\times 18$ matrix over $\mathbb{C}$ with characteristic polynomial equal to
$$
(x-1)^6(x-2)^6(x-3)^6
$$
and a minimal polynomial equal to
$$
(x-1)^4(x-2)^4(x-3)^3.
$$
Assume ...
2
votes
2answers
34 views
Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]
Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?).
How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
0
votes
0answers
11 views
Minor, cofactor of a matrix.
(Minor, Cofactor of a Matrix): The number $ \det \left(A(i\vert j)\right)$ is called the $ (i,j)^{\mbox{th}}$ minor of $ A$ . We write $ A_{ij} = \det \left(A(i\vert j)\right).$ The $ ...
7
votes
3answers
67 views
A rational “fifth root” of the scalar matrix $2I$
I am working on the following problem from a past exam.
Find a necessary and sufficient condition for there to exist a square matrix $A$ of order $n$ whose entries are all rational, such that $A^5 ...
0
votes
1answer
47 views
Intuitive meaning of right and left eigenvector
I am trying to get an intuitive understanding of the meanings of right and left eigenvectors. I guess the best thing you can do is to provide examples of application. (Examples from the field of ...
-3
votes
0answers
17 views
How to use a matrix or matrix algorithm to approximate a function [on hold]
Is it possible to convert a function into a matrix or matrix algorithm
To approximate it
1
vote
1answer
19 views
How to properly sort a set of axis-aligned boxes so they are drawn correctly under this projection?
Given a set S of axis-aligned, non-overlapping boxes {x,y,z,w,h,l}, where x,y,z are their center-positions and w,h,l their width, height and lengths, and given the following orthographic projection:
...
3
votes
0answers
27 views
Bound on the maximum norm of the inverse of an $M$-matrix
I am trying to prove this bound on the maximum absolute row sum norm of the inverse of an $M$-matrix.
Let $A$ be an $M$-matrix and assume $x \in \mathbb{R}^n$ exists such that
$x>0$ ,
...
6
votes
0answers
47 views
Integrals of matrix functions
I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ...
2
votes
3answers
57 views
$A$ and $B$ are rectangular matrices such that $AB$ is invertible, prove that $BA$ isn't.
$A \in \mathbb{R}^{m \times n} $, $B \in \mathbb{R}^{n \times m}$, $m \neq n$
I believe I understand the problem, but I can't come up with a good, formal proof for this..
0
votes
1answer
25 views
Solution space of a system and its rank
If I know that A is an m by n matrix, B is an n by m matrix, rank(A) = rank(B) = rank(AB) = m and that AB = I_m: how can I show that the solution space of Ax=0 is the same as for (BA)x = 0 and how can ...
