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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2
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1answer
16 views

Diagonalizability and elementary divisors

How to prove that an $n \times n$ matrix $A$ over a field $\mathbb F$ is diagonzalizable if and only if every elementary divisor of $A$ has degree $1$? I kind of know why this is true but I am not ...
1
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1answer
58 views

Prove that this $10 \times 10$ matrix is diagonalizable. [on hold]

Suppose that $A$ is an non-invertible $10\times10$ real matrix, and that $\mathrm{rank}(A-3I)=7$ , $\mathrm{rank}(A-I)=4$. How do I prove that $A$ is diagonalizable?
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2answers
28 views

Please prove this matrix is invertible and diagonalise. [on hold]

enter image description here I have no idea with proving the matrix is diagonalise and invertible Thankyou
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2answers
41 views

Finding if a linear transformation is diagonalisable

Hi i am having some trouble tackling this question for my exam revision. Let $M_{(2,2)}(\mathbb{R})$ denote the vector spce of 2x2 matrices over the real numbers. Also, let A denote the matrix ...
1
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1answer
22 views

Find new bases with respect to which the matrix of the linear map is diagonal

Let $U=\mathbb{R}^3$ and $V=\mathbb{R}^3$. Let $T$ be the linear map $U\rightarrow V$ defined by the matrix $$A= \left[ \begin{matrix} 6 &3&9\\ 3&4&2\\ 3&6&0 \end{matrix} ...
1
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2answers
226 views

Prove that an $n \times n$ matrix $A$ over $\mathbb{Z}_2$ is diagonalisable and invertible if and only if $A=I_n$

Through some facts, when $A$ is invertible, I found out that the eigenvalue can't be $0$, since if the eigenvalue is $0$, then $\det(A)=0$, which means that is is not invertible. Since it is over ...
2
votes
2answers
57 views

Diagonalisation proof

Suppose the nth pass through a manufacturing process is modelled by the linear equations $x_n=A^nx_0$, where $x_0$ is the initial state of the system and $$A=\frac{1}{5} \begin{bmatrix} 3 ...
2
votes
2answers
56 views

How to diagonalize a matrix with several eigenvalues of zero?

I'm looking to diagonalize a matrix A seen below. (Find a $P$ and a $D$ such that $AP = PD$). $$ \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 ...
0
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1answer
19 views

Simultaneous Diagonalization of A and B via $\Sigma = A^{-1}B$

I am reading the paper "A Generalized Subspace Approach for Enhancing Speech Corrupted by Colored Noise" by Yi Hu and Philipos C. Loizou. In the paper, they claim that given two matrices $R_{n}$ and ...
1
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1answer
29 views

Given two diagonalizable matrices that commute (AB = BA),is AB necessarily diagonalizable?

Prove or disprove: Given two diagonalizable matrices A, B that commute (AB = BA),is AB necessarily diagonalizable?
1
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3answers
38 views

Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization)

Question: Let $A$ be a $3 \times 3$ Matrix such that $[-3,4,1]$ is the eigenvector corresponding to eigenvalue $3$, and $[6,-3,2]$ is an eigenvector corresponding to the eigenvalue $2$. If $v$ = ...
1
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1answer
38 views

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} \\ ...
1
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2answers
50 views

Is the following matrix diagonalizable?

Determine if this matrix is diagonalizable. $$ C= \begin{bmatrix} \frac{\sqrt2}2 & 0 & -\frac{\sqrt2}2 \\ 0 & 1 & 0 \\ \frac{\sqrt2}2 & 0 & \frac{\sqrt2}2 \end{bmatrix} $$ I ...
1
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3answers
49 views

Is a complex symmetric matrix with positive definite real part diagonalizable?

Let $M \in \mathbb{C}^{n \times n}$ be a complex-symmetric $n \times n$ matrix. That is, $M$ is equal to its own transpose (without conjugation). If the real part of $M$ is positive-definite, then is ...
4
votes
1answer
117 views

Prove there is an orthogonal matrix

Exercise 6.4.15 in Shifrin and Adams' Linear Algebra: a Geometric Approach says Suppose $A$ and $B$ are symmetric and $AB=BA$. Prove there is an orthogonal matrix $Q$ so that both $Q^{-1}AQ$ and ...
0
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1answer
35 views

I'm having a difficulty solving this linear algebra problem. I would appreciate any help!

So we are having a matrix $A$: $$\begin{bmatrix} 2&0&0\\ 0&1&1\\ 0&1&1\\ \end{bmatrix}$$ I already found the eigenvalues which are $\lambda_{1}=2$ (of multiplicity $2$) and ...
2
votes
2answers
37 views

How does diagonalizing a matrix help with eigenvalue calculation?

So in linear algebra class we learned that a matrix $A$ is diagonalizable if it can be written in the form: $$A=PDP^{-1}$$ The useful part was that $A^k$ can be easily computed with $PD^kP^{-1}$. The ...
5
votes
2answers
283 views

Geometric interpretation for eigenvalues and eigenvectors of the cross product's representation as a linear map

Fix ${\bf x} = (x_1,x_2,x_3) \in \Bbb R^3\setminus\{{\bf 0}\}$. We can look at the cross product as a linear map ${\bf x}\times: \Bbb R^3 \to \Bbb R^3$ which is represented in the standard basis by ...
1
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2answers
29 views

If A has complete distinct eigenvalues and A commutes with M and N, do M and N commute?

I've been thinking about the way that eigenvalues appear on the diagonal of a diagonalized matrix, and found a nice question on it in my textbook: Prove that if a matrix $A \in C_n$ with $n$ distinct ...
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0answers
56 views

Unitary Complex Matrix

Let $M$ be a complex matrix $$M:=\begin{bmatrix} 0 & i & -i \\ i & 0 & 1 \\ -i & 1 & 0 \\ \end{bmatrix}$$ 1) Find a unitary complex matrix $Q$ ...
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0answers
21 views

Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form, $C^{-1}AC$

Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form by the transformation $C^{-1}AC$. I know that we are able to diagonalize a matrix and set A = $PDP^{-1}$ where P is the eigenvector ...
0
votes
1answer
28 views

Basis which makes TWO linear transformation diagonalised at once

Find a basis $\gamma$ with respect to which both of the following lienar transformations on $\mathbb{R^3}$ become diagionalised (the matrices below are the matrices with respect to the standard ...
1
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1answer
18 views

Spectral Theorem for a Complex Vector Space and corollary

When it says orthogonal projections it must mean they are orthogonal with each other otherwise if it meant they were orthogonal lin transformations then they would be invertible- the only invertible ...
0
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1answer
29 views

If A is invertible and orthogonally diagonalizable, is $A^{-1}$ orthogonally diagonalizable as well?

I know that the answer is yes. Are the reciprocal of the eigenvalues of A the eigenvalues for $A^{-1}$? If the eigenvalues for A are $3$ and $2$, would the eigenvalues for $A^{-1}$ be $1/3$ and $1/2$? ...
0
votes
1answer
20 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
0
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2answers
57 views

Diagonalizing, Eigenvalues and Eigenspaces

Prove that the matrix $A= \begin{pmatrix} 2 & 0 & -2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix} $$ $ is diagonalizable and thus find the ...
0
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2answers
32 views

Diagonalizing a 3x3 matrix

Prove that matrix $A$ is diagonalizable, find the bases for the eigenspaces, the diagonalizing matrix $P$, and compute $P^{-1} A P$ where $A= \left(\begin{array}{ccc} 2 & 0 & 3 \\ 0 ...
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0answers
20 views

Restriction of diagonalizable endomorphism to an invariant subspace is diagonalizable - another approach

There are some questions discussing the diagonalizability of a restriction of a diagonalizable endomorphism to an invariant subspace, however, I have a question regarding a certain approach, which ...
1
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2answers
26 views

If matrix $A$ is similar to matrix $D$ and $B$ is similar to $E$, than: $AB$ is similar to $DE$?

More specifically: if $A$ & $B$ are diagonalizeable, than is it correct to say that $AB$ is diagonalizeable? (Hints would be more appreciated)
0
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1answer
40 views

eigenvalues and eigenvectors of 2x2 block matrix

My question is a really straightforward one: Is there an easier way to find the eigenvalues and/or eigenvectors of a 2x2 block diagonal matrix other than direct diagonalization of the whole matrix? $ ...
0
votes
1answer
34 views

Eigenvalues and Eigenvectors relating to orthogonal basis and diagonal matrices

Find the eigenvalues and eigenvectors of the matrix. $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1\\ 0 & -1 & 1 \end{bmatrix}$$ As we have seen in the ...
0
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1answer
66 views

Is $A = \left( \begin{matrix} \lambda & 0 \\ 0 & 0 \end{matrix}\right)$ diagonalizable if $\lambda$ is the only eigenvector of $A$?

My book states the following lemma: Suppose that $\lambda$ is the only eigenvalue of $A \in M_{2\times 2}(\mathbb{F})$. Then, $A$ is diagonalizable if and only if $A = \left( \begin{matrix} \lambda ...
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1answer
43 views

How to know if the matrix is diagonalizable?

I put the matrix into a row echelon form but I got the matrix without $a$-s. What should I do ? M= $\pmatrix{1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0}$ It is not OK. I need to ...
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4answers
37 views

Find the limit a matrix raised to $n$ when $n$ goes to infinity

Let $ A $ be a $ 3\times3 $ matrix such that $$A \left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right),~~~A \left( \begin{array}{ccc} ...
1
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2answers
28 views

For which complex parameters the following matrix is diagonalizable

For all possible complex values of the parameter $\lambda$, determine if the matrix $A$ is diagonalizable and if so find an invertible matrix $C$ and a diagonal matrix $D$ so that $C^{-1}$$DC=A$ $A$ ...
2
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1answer
20 views

Does this inner product manipulation make sense?

Suppose A is a normal matrix over $M_n(\mathbb{C})$, with diagonalization $A = PDP^*$. Consider the inner product $<A\mathbf{v}, \mathbf{v}>$ $<A\mathbf{v}, \mathbf{v}> = ...
0
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1answer
25 views

Symmetric Matrices and Diagonalization

Hi, I am trying to figure this problem out, but I am having difficulty. What I do know is that since A is symmetric, then S must be orthogonal. Also that S^(-1) must equal S^t (Transpose). ...
0
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1answer
46 views

Conditions of diagonalizability of $n \times n$ anti-diagonal matrix

Let the matrix $A$ be an anti-diagonal matrix with real number elements where $a_{1n} = \lambda_1$, $a_{2,n-1} = \lambda_2,\ldots, a_{n1} = \lambda_n$. The task is to find out conditions on ...
0
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4answers
75 views

Is the matrix diagonalizable for all values of t?

For t∈R, let $A_t = \left( \begin{array}{ccc} t & 1 & 1 \\ 1 & t & 1 \\ 1 & 1 & t \end{array} \right) $. Find the Eigenvalues and Eigenvectors. Is $A_t$ diagonalizable for all ...
1
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1answer
37 views

Is this matrix diagonalizable, and if so what is it?

I have the following matrix: $$A=\begin{bmatrix} 3 & 0 & 0 \\ 5 & -2 & 0 \\ 0 & 4 & 1\end{bmatrix}$$ Is it diagonalizable? I think it is, but when I try to test the ...
0
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1answer
49 views

What is the significance of being a diagonalizable matrix?

I understand that if a matrix $A$ is diagonalizable, then it is similar to a diagonal matrix $D$. And then the two matrices have the same determinant, rank, and eigenvalues. I am thinking that there ...
1
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1answer
32 views

How to determine if a matrix is diagonizable?

Consider the matrix $A = \begin{bmatrix}3 &0\\ 0& 3\end{bmatrix}$. To determine whether it can be diagonalised, I have found eigvalues $\lambda_1,\lambda_2$ which are both $= 3$, but then I ...
1
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1answer
35 views

Let $A\in M_{n,n}(\mathbb{C})$ be a diagonalisable matrix. Prove $\exists B \in M_{n,n}(\mathbb{C})$ such that $B^{2016} = A$ [duplicate]

Let $A\in M_{n,n}(\mathbb{C})$ be a diagonalisable matrix. Prove $\exists B \in M_{n,n}(\mathbb{C})$ such that $B^{2016} = A$ I can't see why this statement would be true. Perhaps I'm missing ...
1
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1answer
31 views

How can I force least square solution matrix to be diagonal?

Let's say I have the following equation $$AX=B$$ where $A$ is a $8\times 3$ matrix (known), $X$ is a $3\times3$ "diagonal" matrix which represents the coefficients (unknown) and $B$ is a ...
0
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0answers
53 views

Determining if a n by n matrix is diagonalizable with rank < n and given eigenvalues

Question: Let $A$ be a 7 by 7 matrix of rank 5. Assume it is known that 3 and 4 are eigenvalues of $A$. Is A diagonalizable? (Justify your answer.) What I Know: For the corresponding eigenvectors for ...
0
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1answer
28 views

Is there a counter-example where the matrices $A$ and $B$ commutes, $A$ have distinct eigenvalues but $B$ is not diagonalisable?

Suppose that $A$ and $B$ are matrices in $M_n(\mathbb{C})$ with $AB=BA$. We know that if $A$ have $n$ distinct eigenvalues, then every eigenvalue of $A$ is also an eigenvalue for $B$, so $B$ is ...
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3answers
36 views

Exponentiation of Diagonalizable Matrix

Wikipedia says that "If $A = UDU^{−1}$ and D is diagonal, then $e^{A} = Ue^{D}U^{−1}$" Why is this the case? I understand that $e^D$ yields a matrix where $M_{i,j} = e^{D_{i,j}}$, but how is it ...
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1answer
28 views

diagonal times rank 1 matrix still rank 1?

First let me make two statements to give my question the proper context. Consider $D$ a diagonal matrix and $u v^T$ a rank 1 matrix. From my current knowledge there exist computationally cheap ...
1
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1answer
30 views

What does the diagonalized matrix say about a Transformation?

I have a matrix given: $$A=\begin{pmatrix} 7 & -2 \\ -1 & 8 \end{pmatrix} $$ I have found its characteristic polynomial: $\lambda^2 - 15\lambda +54 = 0$, which gave me $\lambda = 6, 9$. Now, ...
2
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0answers
51 views

Finding an eigenbasis for a matrix & diagonalization

I'm trying to find an eigenbasis for matrix A = $\begin{bmatrix}1&-1&1\\-1&1&-1\\1&-1&1\end{bmatrix}$ so that I can use the result to diagonalize A. Because the characteristic ...