2
votes
1answer
106 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
2
votes
1answer
15 views

Factoring a series of Matricies

I have a graduate physics text which has an arbitrary matrix $M$. I am given the following $M^{0} + M^{1} + M^{2}...M^{n-1}$ and then it says this is equal to $(M^{n}-I)(M-I)^{-1}$, where $I$ is the ...
0
votes
1answer
38 views

Factoring a matrix out of linear matrix equation

I'm having a bit of trouble following a solution in a textbook, one step in particular. I have the equation $(Z + tV)^{-1}$ where $Z$, $V$ are matrices and $t$ is a scalar. $Z$ is positive definite, ...
2
votes
2answers
51 views

Factor 9 terms with 3 variables into 4 expression

I just got the determinant from a 4x4 matrix and the simplified version is below. $$ det(M) = \begin{vmatrix} 2k-mw^2 & -k & 0 & 0 \\ -k & 2k-mw^2 & -k & 0 \\ 0 & -k ...
0
votes
1answer
54 views

Frobenius matrix norm vs. 2-norm

From this article about the singular value decomposition: Let $A$ be an $n \times d$ matrix and think of the rows of $A$ as $n$ points in $d$-dimensional space. The Frobenius norm of $A$ is the ...
3
votes
1answer
79 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
1
vote
3answers
65 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
1
vote
1answer
523 views

Fast way to solve a system of linear equations from Givens QR decomposition

I have this system of linear equations: $$ A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix} $$ $$ b= \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} $$ I ...
3
votes
2answers
73 views

Factorization of a linear combination of matrices

I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper) ...
1
vote
1answer
313 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
0
votes
1answer
252 views

Calculating powers of 2 on a 2D grid without factoring.

Consider the following 2D infinitely large grid where the dots represent infinity: ...
1
vote
1answer
1k views

General method for factorizing matrix determinants

I'm learning how to factorize determinants of a square matrix in school, but we haven't learnt a general method to do that, besides 'creating zeros'. So I thought maybe I'll ask here if someone does ...