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 […]

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 […]

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 […]

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 […]

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.

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 […]

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 […]

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 […]

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 […]

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.

Intereting Posts

Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
Uses of step functions
Primitive roots of unity
Integration and differentiation of Fourier series
Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?
Can't prove Continuum Hypothesis
The Dual Pairing
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?
Existence of a normal subgroup with $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$
Given a fixed prime number $p$ and fixed positive integers $a$ and $k$; Find all positive integers $n$ such that $p^k \mid a^n-1$
Exponent of $p$ in the prime factorization of $n!$
Lax-Milgram theorem on Evans. If the mapping is injective why do we need to prove uniqueness again?
Showing the following sequence of functions are uniformly convergent
Factorials and Prime Factors
Uniqueness of Smoothed Corners