Articles of linear algebra

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.

Image of dual map is annihilator of kernel

Suppose $T:V\to W$ and that $V$ is finite-dimensional. I want to prove that $$\text{Im }T’=(\ker T)^0$$ where $T’$ is the dual/transpose map and $(\ker T)^0$ is the annihilator of the kernel. I know that $\phi \in V’$ is an annihilator of $\ker T$ if and only if $$\phi(v)=0 \space \forall v\in \ker T$$ if and […]

How to prove $I-BA$ is invertible

This question already has an answer here: $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate] 3 answers

Direct summand of skew-symmetric and symmetric matrices

Let $W_1$ be the subspace of $\mathcal{M}_{n \times n}$ that consists of all $n \times n$ skew-symmetric matrices with entries from $\mathbb{F}$, and let $W_2$ be the subspace of $\mathcal{M}_{n \times n}$ consisting of all symmetric $n \times n$ matrices. Prove that $\mathcal{M}_{n \times n}(\mathbb{F}) = W_1 \oplus W_2$. I couldn’t really figure out why […]

How to formalize $\text{span}(S)=\{c_1v_1+\cdots+c_kv_k\mid v_1,~\cdots,~v_k\in S,~c_1,~\cdots,~c_k\in F\}$ rigorously in first order language?

In Linear Algebra, the subspace of a vector space $V$ over a field $F$ spanned by a set $S$ of $V$, is defined to be the set of all linear combinations of $S$. Symbolically, it may be written as something like $\text{span}(S)=\{c_1v_1+\cdots+c_kv_k\mid v_1,~\cdots,~v_k\in S,~c_1,~\cdots,~c_k\in F\}$, or $\text{span}(S)=\{c_1v_1+\cdots+c_kv_k\mid\forall i\in\{1,\cdots,k\},~(v_i\in S\wedge c_i\in F)\}$. It is a satisfactory […]

How can I prove that a matrix is area-preserving?

How can I prove that a $2\times2$ matrix $A$ is area-preserving iff $\det(A)=1$ or $\det(A)=-1$?

Suppose p(t)=r(t)s(t) where r(t) is a polynomial with coefficients in the complex numbers. Show r(t) belong to the field of polynomial over R.

Here is the entire problem: “Let $p(t)$ denote a polynomial with real coefficients. Suppose that $p(t)=r(t)s(t)$ where $r(t)$ is a polynomial with coefficients in the complex numbers (i.e $r(t)$ belongs to the set of polynomials over $\Bbb{C}$ and $s(t)$ belongs to the set of polynomials over $\Bbb{R}$). Show that $r(t)$ belongs to the set of […]