Articles of diagonalization

Lagrange diagonalization theorem – what if we omit assumption about the form being symmetric

I know that for every symmetric form $f: U \times U \rightarrow \mathbb{K}$, char$\mathbb{K} \neq 2$ there exists a basis for which $f$’s matrix is diagonal. Could you tell me what happens if we omit assumption about $f$ being symmetric? Could you give me an example of non symmetric bilinear form $f$ which cannot be […]

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not effect which method to use?

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 […]

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3} & \lambda_2^{p_3} & \cdots & \lambda_n^{p_3} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_1^{p_n} & \lambda_2^{p_n} & \cdots & \lambda_n^{p_n} \end{bmatrix}$$ where $p_1=0$ and $p_k > p_i$ for $i < k […]

$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is

$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is Diagonalizable Positive semidefinite $0,3$ are only eigenvalues of $J$ Is positive definite $J$ has minimal polynomial $x(x-3)=0$ so 1, 2,3 are true , am I right?

Show that the linear transformation $T:V\to V$ defined by $T(B)=ABA^{-1}$ is diagonalizable

Notation: If $\Bbb F=\Bbb {R}$ or $\Bbb C$, denote by $M_n(F)$ the $n\times n$ matrices with entries in $\Bbb F$. Let $V=M_3(C)$ be a $9$-dimension vector space over $\Bbb C$ and let $$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1 \\ 0 & 1 & -2 \\ \end{pmatrix}.$$ Define the linear […]

Eigenvalues of a tridiagonal stochastic matrix

I’ve tried to calculate the eigenvalues of this tridiagonal (stochastic) matrix of dimension $n \times n$, but I had some problems to find an explicit form. I only know that 1 is the largest eigenvalue. $$M=\dfrac{1}{2}\begin{pmatrix} 1& 1 & & & & \\ 1& 0 &1 & & & \\ & 1 & 0 &1 […]

Show that if $A^{n}=I$ then $A$ is diagonalizable.

Suppose $A$ is an $m \times m$ matrix which satisfies $A^{n}=1$ for some $n$, then why is $A$ necessarily diagonalizable. Not sure if this is helpful, but here’s my thinking so far: We know that $A$ satisfies $p(x)=x^{n}-1=(x-1)(x^{n-1}+\ldots+x+1)$. If $A=I$ it is clearly diagonalizable so we may assume that $A$ is a root of the […]

Eigenvalue decomposition of $D \, A \, D$ with $A$ symmetric and $D$ diagonal

Let $A$ be a real, symmetric matrix. It admits the eigenvalue decomposition $A = U \Lambda U^T$ where the eigenvectors are chosen to be orthogonal. Let $D$ be a diagonal matrix and $B = D A D = D U \Lambda U^T D = (DU) \Lambda (DU)^T = V \Lambda V^T. \tag{$\ast$}$ Assume none of […]

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that, I have $2$ layers of atoms where $A$ is connectivity within the layer $1$ itself and $B$ is connectivity between layer […]