Articles of diagonalization

Diagonalization of linear operators

First of all, I´m sorry for my English, I´m Spanish so I hope you can all understand me. Here is my problem. Let $T(p(x))=p(x+1)$ be a linear operator from the space of polynomials with real coefficients and degree less than or equal to $n$. I´m having trouble with the matrix of the operator, and without […]

Positive semidefinite but non diagonalizable real matrix – proof real parts of eigenvalues are non-negative

I have a question about positive semidefinite matrices that are non diagonalizable. Example: \begin{equation} A= \left(\begin{array}{cc} 2 & 1\\ 0 & 2\\ \end{array}\right) \end{equation} Clearly the (real part of the) eigenvalues of $A$ are non-negative. But how do I prove in general that the real part of the Eigenvalues of a positive semi-definite real matrix […]

Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues

Let $L:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be defined by $L(A) = A^T,$ where $A^T$ is the transpose of $A$ and $M_n(\mathbb{C})$ is the space of all $n \times n$ matrices with complex entries. Prove that $L$ is diagonalizable and find the eignevalues of $L.$ I worked out a general $L$ that was $2 \times 2$ (might’ve found […]

diagonalize a non-normal matrix , without distinct eigenvalues

I wonder how to diagonalize a matrix that is non-normal, and does not have distinct eigenvalues. Let $\lambda_i$ be the eigenvalue, and $v_i$ be the eigenvector with that eigenvalue. I think the process would go like this: Determine if $\dim(\mathrm{span}(v_i)) = $ multiplicity of $\lambda_i$. If no, then it is not diagonolizable. If yes, go […]

How to determine if a 3×3 matrix is diagonalizable?

The matrix is given as: $A=\begin{bmatrix} 0 & 1 & 1 \\0 & 0 & 4 \\ 0 & 0 & 3 \end{bmatrix}$ So the matrix has eigenvalues of $0$ ,$0$,and $3$. The matrix has a free variable for $x_1$ so there are only $2$ linear independent eigenvectors. So this matrix is not diagonalizable. What […]

Proving every real symmetric matrix is congruent to the canonical form $\mathrm{diag}(\mathbf I,\mathbf{-I},\mathbf0)$

Let $\bf A$ be a real symmetric matrix of order $n\times n$ such that $\mathrm R(\mathbf A)=r(\le n)$. Then show that there exists a nonsingular matrix $\bf F$ such that $\bf F’AF=\begin{pmatrix}\bf I & \bf0 & \bf0 \\\bf0 & \bf-I & \bf0 \\\bf0 & \bf0 & \bf0 \\\end{pmatrix}$ where the orders of $\bf I$ and […]

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 […]