Articles of linear algebra

unique factorization of matrices

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 … X_n$ where $X_i \in U$ and this factorization is unique? We may assume the […]

Priority vector and eigenvectors – AHP method

I’m reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each alternative over every other alternative, what meaning do we attach to the vector obtained by weighting the preferences by the corresponding priorities of the […]

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I’m given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in $\mathbb{C}^3$. What I’m having trouble showing is that mapping $g$ to $h(g)$ is a group homomorphism. Namely, I know that $h(ab)p(z)=p(b^{-1}a^{-1}z)$, but I can’t […]

Show the negative-definiteness of a squared Riemannian metric

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices. The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric $$ d(A,B):= \Bigg(\operatorname{tr}\bigg(\ln^2\big(\sqrt{A^{-1}}B\sqrt{A^{-1}}\big)\bigg)\Bigg)^{\frac{1}{2}}. $$ EDIT It could also be defined as follows $$ d(A,B):= \lVert\log(A^{-1}B)\rVert_{F} $$ EDIT II I could show the negative-definiteness of $d^2$ if I could write […]

Is there a classic Matrix Algebra reference?

I’m looking for a classic matrix algebra reference, either introductory or advanced. In fact, I’m looking for ways to factorize elements of a matrix, and its appropriate determinant implications. Your help is greatly appreciated.

Linear dependence in a complex vector space, and viewed as a real vector space

Suppose $M$ is a linearly dependent set in a complex vector space $X$, Is $M$ linearly dependent in $X$, regarded as a real vector space?? My attempt Say $dim X = n$ regarded a comlex vector space. We know number of vectors in $M$ is $<$ than $n$. since $M$ is linearly dependent. We know […]

Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: Give a transition matrix $M$ such that there exists no invertible matrix $U$ with $U^{-1} M U$ a diagonal matrix. Is there a combinatorial […]

If $A \in M_{n \times 1} (\mathbb K)$, then $AA^t$ is diagonalizable.

Let $A \in M_{n \times 1} (\mathbb K)$. I’m asked to proof that $AA^t$ is diagonalizable. My attempt: If $A = 0, \, AA^t = 0$ is diagonal. Let $A = \begin{bmatrix} a_1\\\vdots \\ a_n \end{bmatrix} \neq 0$, then $AA^t = \begin{bmatrix} a_1\\\vdots \\ a_n \end{bmatrix} \begin{bmatrix} a_1&… & a_n \end{bmatrix} = \begin{bmatrix} a_1 a_1 […]

What's special about the first vector

My linear algebra notes state the following lemma: If $(v_1, …,v_m)$ is linearly dependent in $V$ and $v_1 \neq 0$ then there exists $j \in \{2,…,m\}$ such that $v_j \in span(v_1,…,v_{j-1})$ where $(…)$ denotes an ordered list. But if at least one $v_i$ is $\neq 0$ then the list can be reordered and the lemma […]

symmetric positive definite matrix question

Let $A$ be an $n\times n$ symmetric positive definite matrix and let $B$ be an $m\times n$ matrix with $\mathrm{rank}(B)= m$. Show that $BAB'$ is symmetric positive definite.