# Questions tagged [diagonalization]

For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

1,779 questions
Filter by
Sorted by
Tagged with
20 views

### Finding a diagonal matrix $B$ and a unitary matrix $C$ that satisfy $B=C^{-1}AC$.

The matrix $A$ is given as $$A=\frac{1}{9} \begin{bmatrix} 4+3i & 4i & -6-2i \\ -4i & 4-3i & -2-6i \\ 6+2i & -2-6i & 1 \end{bmatrix}$$ Find a diagonal matrix $B$ and a ...
18 views

24 views

11 views

26 views

### Unitary matrix U to diagonalize matrix A

I'm working on this exercise and I got stuck. Find a unitary matrix $U$ and a diagonal matrix $D$ such that $A=U^{*}DU$ for $$A=\begin{pmatrix}1&i\\-i&1\end{pmatrix}$$ So what I decided to ...
25 views

38 views

### A problem of showing a matrix is diagonal.

$\mathbf {The \ Problem \ is}:$ Suppose, $A$ be a $2×2$ real matrix with $tr(A) =0$ and $det(A) =-1.$ Then show that : $(a)$ $\mathbb R^2$ has a basis consisting of eigenvectors of $A .$ $(b)$ ...
31 views

### Prove that a matrix is not diagonalizable for any scalars $a$, $b$.

I need to prove that \begin{align*} A= \begin{bmatrix} a & 1 & 0\\ 0 & a & 0\\ 0 & 0 & b\\ \end{bmatrix} \end{align*} is not diagonalizable for any scalars $a$ and $b$. I've ...
20 views

### What does Canonical Form Means Intuitively?

I have seen multiple times where canonical form is mentioned. I went to Wikipedia and as usual its quite formal definition and not intuitive at all. So the following context taken from MathWorld, what ...
75 views

### How to find $A^5$ in matrix $A$ with eigenvalues and eigenvectors given

Let $A$ be a $3 \times 3$ diagonalizable matrix whose eigenvalues are $\lambda_1=2, \lambda_2=4$, and $\lambda_3=3$. If $$v_1=[1, 0, 0], v_2=[1, 1, 0], v_3=[0, 1, 1]$$ are eigenvectors of $A$ ...
24 views

### Are there operators on arbitrarily large vector spaces with no eigenvalues?

So, we know that operators (linear transformations, square matrices, whatever you wanna call 'em) on complex vector spaces always have an eigenvalue. Moreover, operators on real vector spaces of odd ...
50 views

### Determine an invertible matrix $S \in \operatorname{Mat}_3(\mathbb{R})$ such that $S^{-1}AS$ is a diagonal matrix

Consider the matrix $$A = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 2 & 2 \\ -1 & 0 & 1 \end{pmatrix} \in \text{Mat}_3(\mathbb{R})$$ Then I have to determine an invertible matrix ...
42 views

### Showing if all eigenvalues of $A$ have negative real parts then our system has a strong Lyapunov function of the form $x^TSx$.

Can I please have help solving the problem? I am having a tough time working out the details for $S$ and how to do it without assuming diagonability. Thank you! Show that if all eigenvalues of $A$ ...
58 views

25 views

### Diagonalize an unknown matrix given only its eigenvectors and eigenvalues

The question is the following: The only way I can think of in doing this question would be for me to set variables for all values of A, and then using the given Eigenvectors and values to solve for ...
31 views

### Crucial missing step in proof requires simpler justification

Let $p+q+r=1, p, q, r > 0.$ Let $M = \begin{bmatrix}p&q&r\\ r&p&q\\q&r&p\end{bmatrix} = PDP^{-1}$ where $D = \text{diag}(1, \alpha, \beta)$ (this is shown using Perron-...
22 views

### Diagonalizability of multi-dimensional mass-spring-damper state-transition matrix

It is well known the differential equation related to a single degree of freedom mass-spring-damper system, $$m \ddot{x} + d \dot{x} + k x = 0 ,$$ where, $m > 0$ , $d > 0$ and $k > 0$ , ...
I have the equation bellow which I want to solve for $I$ with constraints on $I$. where $I$ is a vector and $lb \leq I_{i} \leq ub$ . Note that $size(I)=(N,1)$ , $size(A_{m})=(N',m)$ , $size(K)... 1answer 58 views ### Every square matrix is a sum of two diagonalisable matrices I've been stuck with this question for quite a while and am not sure where to start: Prove that if$A$is an$n \times n$matrix, then$A$can be written as$B + C$where both$B$and$C$have$n\$ ...
given an Hermitian matrix $$A = A^{\dagger}$$ is it always true that there wil exist another matrix (unitary) so we have always that $$PAP^{\dagger}=D$$ where D is a diagonal matrix is it true ...