For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

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2answers
21 views

Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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1answer
24 views

Linear Algebra 3x3 matrix diagonalization Row operation before inversing

Hello I am diagonalizing the matrix $$\begin{bmatrix} -1 & 2 & 2 \\ 2 & 2 & -1 \\ 2 & -1 & 2 \end{bmatrix}.$$ The eigenvalues I found are $-3$ and $3$. The eigenvectors are ...
2
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2answers
44 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
0
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1answer
17 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. But I don't know how to prove, that in the similarity ...
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1answer
28 views

Three Simultaneously Diagonalizable Matrices

I have three symmetric square matrices $M$, $G$, and $S$ with the following properties: $S$: symmetric and positive semi-definite. $M$: Fully diagonal with positive entries. $G$: is a subset of ...
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2answers
17 views

Similar matrices C and D: how to derive the relation $\mathbf{x} = S^{-1} \mathbf{y}$ when $C = S^{-1}DS$

D (with corresponding eigenvector $\mathbf{x}$) and C (with corresponding $\mathbf{y}$) are similar matrices, which means they have the same eigenvalues. So the relation $C = S^{-1}DS$ holds. So we ...
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4answers
241 views

Exercise about Matrix diagonalization

Well I have to diagonalize this matrix : $$ \begin{pmatrix} 5 & 0 & -1 \\ 1 & 4 & -1 \\ -1 & 0 & 5 \end{pmatrix} $$ I find the polynome witch is $P=-(\lambda-4)^2(\lambda-6)$ ...
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2answers
38 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
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1answer
40 views

Shortcut when finding D when diagonalizing matrices when encountered with tedious matrices

P is given as P = $\left(\begin{array}{rrr} 1 & 1 & 1\\ 1 & 0 & -2\\ 1 & -1 & 1 \end{array}\right).$ It is known that P is invertible. I is a 3x3 identity matrix Supposed ...
0
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2answers
31 views

Show a matrix is similar to a lower triangular matrix

$A = \left(\begin{array}{cc}2 & -1 \\0 & 2\end{array}\right)$ $B = \left(\begin{array}{cc}\lambda & 0 \\1 & \lambda\end{array}\right)$. I know that the $\lambda = 2$. And $r(1,0)^t$, ...
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2answers
28 views

Finding a non-singular matrix $C$ such that $C^{-1}AC$ is diagonal

$A = \left(\begin{array}{cc}1 & 0 \\1 & 3\end{array}\right)$. I find the eigenvalues = 1,3. The eigenvector corresponding to 1 = $t(1,-2)^t$. The eigenvector corresponding to 3 = $r(1,0)$. ...
0
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0answers
21 views

Find a matrix $P$ for a square matrix $B$ with all entries $(B)_{ij} = b$, $b \in R$. $P$ is a matrix that orthogonally diagonalize matrix $B$.

The condition must meet $$D = (P^{T})BP $$ or $$D=(P^{-1})BP$$ I'm having trouble finding a pattern for all entries and infinite square size matrix. I found a matrix P that is 5x5 with all ...
0
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1answer
24 views

Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
0
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0answers
9 views

Dimensions of matrices around a diagonal matrix?

The matrices $\mathbf{L}$, $\beta$ and $\mathbf{c}$ are ($j \times b$), ($b \times 1$) and ($j \times 1$) dimensional, respectively, with $j \le b$. The matrix $\mathbf{X}' \mathbf{X}$ is a diagonal ...
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1answer
42 views

Understanding Jordan Canonical Form.

Two questions: How does the nilpotent index $k$ of a linear transformation L on a vector space of dimension $n$ relate to possible Jordan Canonical Forms? My understanding is that a Jordan block ...
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0answers
29 views

Does gaussian elimination always work?

If so, why don't we use that to get from any square matrix to a triangular matrix - from which can be deduced eigenvalues, determinant (product of eigenvalues) and diagonal matrix (since the diagonal ...
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1answer
24 views

Diagonalization: How to show that A exists $S^2 = D$ given that D is a nonegative diagonal marix

(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such that $S^2 =D$ SOLVED (b) Show that if A is a diagonalizable matrix with ...
0
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0answers
19 views

How to Solve: Find bases that Diagonalize a Matrix

Given bases, $w_1 = (1,0)$ and $w_2 = (0,1)$. Find bases $(e_1,e_2,e_3)$ in $R^3$ relative to which the matrix of $T$ = $ \left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & ...
0
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2answers
75 views

If $A^3 = A$ then the eigen values are all 1 right?

Since $A^n = PD^nP^{-1}$ where D is a matrix consisting only of the eigenvalues of on its leading diagonal. For the scenario to be true $D^B = D$ which is only true if the eigenvalues are all 1s ...
0
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2answers
35 views

Diagonalization: Am i finding these eigenvectors wrongly?

$$A=\begin{bmatrix} 1&-2&-8\\ 0&-1&0\\ 0&0&-1 \end{bmatrix}$$ $$P=\begin{bmatrix} 1&-4&1\\ 1&0&0\\ 0&1&0 \end{bmatrix}$$ Confirm that P diagonalizes A. ...
2
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3answers
44 views

Using factorisation, $A=PJP^-1$ to compute $A^k$

Using factorisation, $A=PJP^{-1}$ to compute $A^k$, where $k$ represents an arbitrary positive integer. $$ \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{-1} & \mathbf{2} \end{bmatrix} = ...
0
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2answers
33 views

A,B are diagonalizable matrix and their characteristic polynomials are the same.prove that $A$ and $B$ are similar

let A,B are diagonalizable matrix in ${c^n}$ and their characteristic polynomials are the same. can we prove that $A$ and $B$ are similar?
2
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2answers
82 views

Conditions for a Matrix to be Diagonalizable

Let $M$ be a matrix with the entries $a_{1}, ..., a_{n}$ on the secondary diagonal (the one that ranges from $m_{n1}$ to $m_{1n}$) with all other entries being $0$. Find under which conditions the ...
2
votes
2answers
141 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
0
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3answers
60 views

Matrix diagonalization - eigenvalues on diagonal

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $\Delta$ such that $A=PD P^{-1}$ where $D$ is a diagonal matrix. What theorem tells us that $P$ is a matrix composed of the ...
3
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4answers
162 views

Showing when a permutation matrix is diagonizable over $\mathbb R$ and over $\mathbb C$

For a permutation $\sigma$ of the set $\{1,...,n\}$, and consider the $n \times n$ matrix $A_\sigma$, where the $i^{\text{th}}$ column is the standard vector $e_{\sigma (i)}$. For which $\sigma$ is ...
1
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2answers
37 views

powers of a diagonal matrix to infinity

Let $A$ be a square matrix that is diagonalizable. This means that it can be like this: $A = SDS^{-1}$, where $D$ is a diagonal matrix containing the eigenvalues of $A$. It follows $S$ contains the ...
2
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2answers
80 views

Let $A$ be a complex matrix such that $A^n = I$, show that $A$ is diagonalisable.

Let $A$ be a complex matrix such that $A^n = I$, show that $A$ is diagonalisable. How do I do this? I would hazard a guess that since $A^n = I$ that means that $A^n$ is obvious diagonal, and I ...
0
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1answer
24 views

Diagonalization: if $P^{-1} A P = \Sigma$, then $P^{-1}\Sigma P = A$?

I am learning about diagonalizing a matrix, one of the theorem is that given $P$ some nonsingular matrix, then we can find a $\Sigma$ such that $P^{-1} \Sigma P$ = $A$, where $\Sigma$ is pure ...
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1answer
22 views

Existence of Matrix

Given endomorphism $\varphi : V \rightarrow V$ of 2 dimensional linear space V and two non-zero vectors $\alpha, \beta \in V$ are given. Assume that $\varphi(\alpha)=4 \alpha , \varphi (\beta) = 3 ...
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3answers
28 views

Linear algebra - eigenvectors of $B$ not diagonalizable.

Explain why the eigenvectors of $B$ does not constitute a basis for $\Bbb R^3$ and $B$ are thus not diagonalizable. $B= \left ( \begin{matrix}2 & 0 & 0\\ 1 & 2 & -1 \\ 0 & 0 & ...
2
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0answers
15 views

Normal form over $\mathbb{Z}$ of matrices of order $2$

Suppose $M \in GL_k(\mathbb{Z})$ is of order $2$. That is, $M^2 = 1$ and $M \ne 1$. Then is it true that upto a change of $\mathbb{Z}$ basis, $M$ has the form $$\begin{pmatrix}J \\ & J \\ & ...
2
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1answer
49 views

Diagonalize a Hermitian matrix with a constraint on the unitary transformation.

I have a $2n\times 2n$ Hermitian matrix $H$ which I want to diagonalize, with the requirement that the unitary transformation be of the form: $$ U = \left( \begin{array}{c c} W & V \\ V^* & ...
1
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2answers
50 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
1
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1answer
38 views

Symmetric positive semidefinite matrix is the square of a symmetric matrix

I am trying to show that matrix $A$ is symmetric positive semidefinite if and only if there exists a symmetric matrix $B$ such that $B^2 = A$. Here is my solution, any comments? I have attempted ...
3
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1answer
55 views

A square matrix with the diagonal and antidiagonal elements different from zero. Looking for some already known property.

I am interested in the properties of a matrix with elements different from zero only on the main diagonal and antidiagonal, like this: $$ \begin{matrix} a & 0 & 0 & h \\ ...
0
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2answers
109 views

Find bases so that Transformation matrix is diagonal

Find bases $(e_1,e_2,e_3)$ and $w_1,w_2$ in $R^2$ relative to which the matrix of $T$ = $ \left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$ is in diagonal form. Ans: ...
3
votes
2answers
56 views

If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$

If $A,B$ are square matrices and $A^2=A,B^2=B,AB=BA$, then calculate $\det (A-B)$. My solution: consider $(A-B)^3=A^3-3A^2B+3AB^2-B^3=A^3-B^3=A-B$, then $\det(A-B)=0\vee 1\vee -1$ The result of ...
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0answers
16 views

Determine the center of $\text{GL}_n(\mathbb{R})$ [duplicate]

Determine the center of $\text{GL}_n(\mathbb{R})$ Where $\text{GL}_n(\mathbb{R})$ is an algebraic group of all $n \times n$ invertible matrices. The center is the subset of those matrices that are ...
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1answer
31 views

Largest positive eigenvalue of a matrix

I am dealing with the Capacity of constrained noiseless communication channels. It has been said that the channel capacity of such a channel is $\log{\lambda}$, which $\lambda$ is the largest positive ...
1
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1answer
38 views

Any good books to practice on Endomorphisms as related to Diagonalization, Cayley-Hamilton, etc.?

Well, I am looking for books (graduate level) that covers linear maps (endomorphisms, to be specific) with emphasis on topics related to numerical linear algebra, like: diagonilzation, ...
3
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2answers
24 views

For an endomorphism of $rank=1$, prove that $Im(u)\subset Ker(u)$ iff $u$ is NOT diagonilsable.

Well, I got this question that I couldn't solve: Problem: Let $E$ be a vector space over $R$ of finite dimension, and $u$ and an endomorphism of $E$ of rank 1. Prove that $$Im(u)\subset Ker(u)$$ if ...
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0answers
30 views

Is this matrix with SVD diagonalizable

Let $X=U\Sigma V^T$ is an (economical) SVD decompoisition of a square $n \times n$ stochastic matrix $X$, where $U$ and $V$ are two $n \times r$ matrices, and $\Sigma$ is a $r \times r$ matrix. Now ...
2
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2answers
40 views

Is it true that if $A$ is unitary diagonalizable and $B$ is similar to $A$ then $B$ is also unitary diagonalizable?

Is it true that if $A$ is unitary diagonalizable (i.e. $A$ is normal) and $B$ is similar to $A$ then $B$ is also unitary diagonalizable (i.e. $B$ is normal) ? Here are my thoughts: If $A$ is unitary ...
2
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1answer
41 views

If $A$ is a real unitary matrix then $A$ is similar to $A^*$

I'm trying to prove the following statement : If $A$ is a real unitary matrix then $A$ is similar to $A^*$. Here is what I have so far: $A$ is unitary and therefore is unitary diagonalizable. Thus, ...
1
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1answer
39 views

Silly question: Why is it in the diagonalization process we can choose our eigenvector to be anything we want?

Suppose we want to diagonalize the matrix $A = \begin{bmatrix}0& 1\\ -1& 0\end{bmatrix}$ Then using $det(\lambda I - A) = 0$ We find our eigenvalues to be $\lambda_{1,2} = \pm j$ Solving ...
3
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1answer
33 views

How does the laplace transform diagonalize the derivative operator?

I was reading this post here and I got really confused at the part where the claim is that the laplace transform diagonalize the derivative operator ...
5
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0answers
44 views

Why is it useful to know when a linear operator on a vector space is diagonalizable?

I'm currently taking a conceptual course in linear algebra, and I'm trying to understand why it would be theoretically useful or illuminating to know when a linear operator (or its matrix ...
0
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1answer
46 views

Diagonlisation of certain matrices

Why is it that the matrix $\begin{pmatrix} -1 & -3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{pmatrix}$ is diagonalisable, even though it has eigenvalues 1, 2, 2 (which are not all ...
1
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0answers
72 views

Determine all $t\in\mathbb{R}$ for which $A_t$ is diagonalizable.

I have this matrix: $$A_t=\begin{pmatrix}\phantom{-}2+t&\phantom{-}4&\phantom{-}2+t&\phantom{-}2+t\\\phantom{-}t-2&\phantom{-}0& -6+t& ...