All Questions
Tagged with characteristic-polynomial diagonalization
24 questions
0
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32
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What do I do when the first column of my characteristic equation for finding eigenvectors are zero?
$$A=\begin{pmatrix} 2 &4 &1\\
6 &5 &2\\
3& 1& 0\end{pmatrix}$$
I got the eigenvalues to be $-2,3,-1$ and I've gotten the eigenvectors of the first $2$ eigenvalues. I'...
-3
votes
1
answer
49
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Finding characteristic polynomial of a square matrix and how to proving that matrix is diagonalizable [closed]
Let $A$ be a square matrix of order $n$ such that $|A + I| = |A − 3I| = 0$ and also $\operatorname{rank}(A)= 2$. I need to find characteristic polynomial of $A$ and have prove that $A$ is ...
1
vote
1
answer
104
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Given Ker(T) and Ker(T-2I), is T diagonalizable?
I'm stuck on this problem that I found on my book of linear algebra.
Be $T:\mathbb{R}ˆ3 \rightarrow \mathbb{R}ˆ3$ a linear map such that
$$Ker(T-2I) = \{(x,y,z) | x+y=0\} $$ and $$Ker(T) = \{ (x,y,z)|...
- 11
0
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1
answer
153
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Showing $\lambda I_V$ diagonalizable and has only one eigenvalue
Problem:
Let $V$ be a finite-dimensional vector space, and let $\lambda$ be any scalar. For any ordered basis $\beta$ for $V$, prove that $[\lambda I_V]_{\beta}=\lambda I$. Then compute the ...
user1055322
0
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0
answers
37
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The eigenspaces of a matrix
I have a list of vectors $v_1, \cdots, v_m \in \mathbb{R}^n$ with $v_i \not=0$ and i'm trying to better understand the eigenspaces of the matrix $M = v_1 {v_1}^T + \cdots + v_m v_m^T$, which is ...
0
votes
1
answer
113
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how to solve this third degree characteristic polynomial?
(this exercise is like the previous question I've written today, but now I have a 3-by-3 matrix (with a real parameter $k$). I need to say where the matrix could be diagonalized, if it's possible. (...
2
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0
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43
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Prove $n\times n$ matrix with uniform random elements is diagonizable?
Let $X$ be a $n\times n$ matrix with $n\in\mathbb{N}^{+}$ and all off diagonal elements of $X$ following a uniform distribution $x_{ij}\sim\mathcal{U}(0,1)$ for any $i,j\in\{1,...,n\}$ with $i\neq j$. ...
user895064
0
votes
1
answer
92
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Are the matrices diagonalizable in the field $K$?
I want to check if the following matrices are diagonalizable in the field $K$.
(a) $A=\begin{pmatrix}2 &1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{pmatrix}, \ K=\mathbb{C}$
(b) $A=\...
- 14.1k
1
vote
1
answer
46
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$n× n$ matrix being non diagonalisable.
If you let $n$ be greater than or equal to $2$ and let $M$ be an $n×n$ matrix with minimal and characteristic polynomial equal to $(x-a)^n$ for some $a$ in the reals. Why would this not be ...
- 11
1
vote
2
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40
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Diagonalization of a particular matrix
I want to prove the following fact: let $A$ be a non-zero square matrix (matrix of an endomorphism in some basis) whose column vectors are the same, i.e. $A = \begin{pmatrix} a_1 & \dots & ...
- 533
2
votes
1
answer
66
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rank of $E_i$ in the sum $A=\lambda_1E_1+\dots+\lambda_kE_k$ where $\lambda_i$ are all the eigenvalues of A.
I'm trying to solve the following exersice.
Let $A$ be a $\nu\times\nu$ matrix with elements over a field $F$ and let $\chi_A(x)=(-1)^\nu(\lambda_1-x)^{\sigma_1}\dots(\lambda_k-x)^{\sigma_k}$ be the ...
- 681
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2
answers
1k
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$A$ is diagonalizable if the minimal and characteristic polynomial of $A$ are equal.
Problem: The square real matrix $A$ is diagonalizable if the minimal and characteristic polynomial of $A$ are equal.
I think I've got this figured out if I can say that the characteristic polynomial ...
user714860
2
votes
1
answer
24
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Finding the power of a complex valued matrix with only the characteristic polynomial
Hi I have this question here.
Suppose that $A \in M_4(\mathbb{C})$ has the characteristic polynomial $$p_A(x)=(x^2+3)(x^2+\sqrt{3}x+3)$$ Find $A^{12}$.
I get my eigenvalues are $\frac{-\sqrt{3}+3i}{...
- 2,917
0
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4
answers
364
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Exercise on expression with matrices using Cayley-Hamilton theorem
For the following matrix $$A=
\begin{bmatrix}
0 & 1 & 0\\
1 & 0 & 1\\
0 & 1 & 1
\end{bmatrix}
$$
I need to use the Cayley-Hamilton theorem to calculate $(A+I_3)^{10}(A-I_3)^2+A$...
user812250
1
vote
1
answer
74
views
For which integer values of n is the matrix diagonalizable by a real matrix?
I need to find the values of n for which matrix C is diagonalizable by a real matrix.
$$ C=\begin{pmatrix}
1 & 1\\
n & n + 1\\
\end{pmatrix}
$$
I've calculated the characteristic polynomial $...
- 11
0
votes
1
answer
145
views
Proving a matrix is diagonalizable given the characteristic polynomial.
If we are given a real symmetric 2x2 matrix how can we deduce that it is diagonalizable given that we know the characteristic polynomial?
- 1
6
votes
1
answer
493
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If $A,B$ are diagonalisable, does $AB$ diagonalisable imply $BA$ diagonalisable?
As discussed in this other post, $AB$ and $BA$ always have the same characteristic polynomial, but not necessarily the same minimal polynomial.
This means that $AB$ diagonalisable does not imply $BA$ ...
- 7,653
0
votes
4
answers
35
views
Characteristic polynomial in $\mathbb{C}^2$
Consider a finite-dimensional vector space $V$ over a field $\mathbb{F}$.Let $a,b\in\mathbb{F}$. State whether the following is true or not: If $\begin{pmatrix} b & a \\ 0 & b \end{pmatrix}$ ...
user731634
-1
votes
2
answers
68
views
diagonalize the following n×n matrix. I am wondering if my solution for the characteristic polynomial is valid or if there is a better way to do it.
The following is the question:
The following is my answer.
1
vote
2
answers
81
views
Diagonalizability of a matrix
Show that $$ A :=\begin{pmatrix} 1 & 0 & 0 \\ -2 & 1 & 2 \\ -2 & 0 & 3 \end{pmatrix}$$ is diagonalizable.
What I did:
First, I determined the characteristic polynomial $$\...
- 635
0
votes
1
answer
172
views
Drawing conclusions from a characteristic polynomial
I was given a characteristic polynomial and was asked to draw some conclusions from it, but i need some help with figuring something out. My polynomial is this : $\lambda^4 -3\lambda^3 +\lambda^2 +3\...
1
vote
3
answers
663
views
Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$, for what values of $a$ given matrix must be diagonalizable.
Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$. For what values of $a$ given matrix must be diagonalizable.
I am talking about diagonalizability over reals.
Efforts:
If a ...
- 5,581
2
votes
4
answers
957
views
If square matrix A satisfying $A^2-4A+4I=0$ does it follow that A is diagonizable?
I am given the following statement and asked to determine whether it is true or false:
If $A$ is an $n\times n$-matrix, and $A^2-4A+4I=0$, then $A$ is diagonizable.
Any help is appreciated, thank ...
- 497
1
vote
3
answers
204
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Diagonalizing a particular $3\times3$-matrix.
Trying to diagonalize the matrix:
$$A=\begin{pmatrix}2 & -\frac{1}{2} & -\frac{1}{2} \\-\frac{1}{2} & 3 & -\frac{1}
{2} \\-\frac{1}{2} & -\frac{1}{2} & 5\end{pmatrix},$$
I ...
- 697