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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1answer
25 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
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1answer
16 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 ...
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2answers
52 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 ...
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2answers
22 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
18 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 ...
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1answer
37 views

How to find a diagonalizable linear map T such that any subspace W is T-invariant but not T*-invariant? [closed]

Let V be a finite-dimensional complex inner product space. Let $W$ be a subspace of $V$ not equal to $\{0\}$ or $V$. Construct a linear operator $T$ on $V$ such that (i) $T$ is diagonalizable (ii) ...
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2answers
25 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)
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1answer
31 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
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1answer
30 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
64 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
41 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
36 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} ...
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2answers
25 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
24 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
34 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 ...
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4answers
73 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
34 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
30 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 ...
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0answers
50 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
23 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
31 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 ...
1
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1answer
26 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
37 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 ...
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2answers
38 views

Solving a system of differential equations using diagonalization

Solve the system \begin{align*} y'_1&= \phantom{-2}y_1 \\ y'_2&= -2y_1-4y_2 \end{align*} I think they want me to solve it by using diagonalization. So far so good. I got the following: The ...
2
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1answer
34 views

Requirements for Diagonal Lemma

What are the axioms required for a formal system to be able to state the Diagonal Lemma or Fixed Point Theorem? If possible, could you please also relate with the following systems, if it applies to ...
0
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1answer
81 views

Show A is diagonalizable if and only if A is similar to a diagonal matrix.

I'm having trouble with the second direction of the proof, ie. Assuming $A$ is similarto a diagonal matrix, then I want to show that $A$ is diagonalizable. I know part (b) of my answer is incomplete, ...
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0answers
26 views

Proving $aI+A$ is Positive Definite

Let $A \in M_{n \times n}^\Bbb C$ be a self adjoint matrix. Prove that there is $a \in \Bbb R$ such that $aI+A$ is a positive definite matrix. What I did so far Let $v$ be a vector in an ...
2
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3answers
104 views

When is matrix $A$ diagonalizable?

I got the following matrix: $$ A = \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix} $$ I need to answer when this ...
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0answers
24 views

Ordering of eigenvectors to maximise trace of diagonalising matrix

Suppose $H$ is a Hermitian matrix. Then it may diagonalised by a unitary matrix $U$, such that $D = U^\dagger H U$ where $D$ is diagonal with elements the eigenvalues of $H$. We may construct $U$ by ...
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0answers
55 views

Is the converse to this theorem true?

In the book that I'm reading there is this one theorem which states. Let V be a finite-dimensional vector space over a field F not of characteristic two. Then every symmetric bilinear form on V is ...
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0answers
28 views

Using Lagrange polynomials to write a matrix

Good evening, I am currently going through some of the exercises in Linear Algebra $2$nd edition by Hoffman and I have come across a question that I don't know how to solve. $T$ is the diagonizable ...
4
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1answer
72 views

Eigenvalues of the Sum of a Positive Definite Diagonal Matrix and a Rank $2$ Skew Symmetric Matrix

Consider a $N \times N$ matrix $A=B_1+B_2$, where $B_1$ is a diagonal matrix with all the diagonal entries between $0$ and $1$ and $B_2$ is a skew symmetric matrix which can be written as $$B_2= ...
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0answers
7 views

How to Define a Hyper Diagonal?

Assume you have a data which dimensions are 777:1. If you take any data of it from the interval 1:100 and express it later with FFT and you get a hyper diagonal in the frequency presentation. If you ...
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2answers
65 views

Simultaneous Diagonalization of two bilinear forms

I need to diagonalize this two bilinear forms in the same basis (such that $f=I$ and $g$=diagonal matrix): $f(x,y,z)=x^2+y^2+z^2+xy-yz $ $g(x,y,z)=y^2-4xy+8xz+4yz$ I know that it is possible ...
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0answers
24 views

Proof of the diagonalization of the probability matrix of the sum of two binomial distributions

I am analysing a statiscal problem where a vector $X\in\mathbb Z_{\geq 0}^{n+1}$ is probabilistically transformed according to $x_i \mapsto \mathrm{Binomial}(x_i, 1/2+\epsilon/2) + ...
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3answers
40 views

How to show $A$ and $A^T$ have same eigenvalues if $A$ is a square matrix?

Is there a way to show it through determinant way rather than standard equation $T(x)=\lambda x$? For instance $\det (A-\lambda I)=\det(A^T-\lambda I)$. I don't know how to correctly express the ...
1
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1answer
32 views

If $A \in \mathbb{R}^{m\times n}$, $m<n$, and $AA^T = I_m$, what are the eigenvalues of $3I_n - A^TA$?

If $A \in \mathbb{R}^{m\times n}$, $m<n$, and $AA^T = I_m$, what are the eigenvalues of $3I_n - A^TA$? How can I start? I know the eigenvalue of $I$ is $1$, but how do I find the eigenvalues ...
1
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1answer
36 views

Numerical diagonalization of a random hermitian matrix $H=U\Lambda U^{-1}$: enforce uniqueness and uniformity of $U$

I've stumbled across this seemingly simple question, but I could not find a satisfactory answer. Suppose I have a complex hermitian random matrix $H$. It can be diagonalized by a unitary ...
0
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1answer
30 views

Matrix $A$ has an eigenvalue with multiplicity $>1$, is $A$ diagonalisable?

Matrix $A$ has an eigenvalue with multiplicity $>1$, is $A$ diagonalisable? I know that if $A$ has distinct eigenvalues $\Rightarrow$ all eigenvectors are linearly independent $\Rightarrow$ I can ...
0
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1answer
42 views

Showing that two matrices are similar by showing they are similar to the same diagonal matrix

The exercise gives two matrices $A$ and $B$ and asks you to show they are similar by showing that they are similar to the same diagonal matrix, and then after that find and invertible matrix $P$ such ...
0
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1answer
34 views

Property of multiplation of a diagonal matrix

I have the singular value decomposition of an image: $X = U \Sigma V^T$, where $\Sigma$ is diagonal matrix. I want to reformulate this equation like this: $X=\Sigma D$ where $D$ is any combination of ...
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2answers
49 views

How to show matrix $A$ diagonalizable iff $A^k$ is diagonalizable for $k\ge 2$?

In order for a matrix to be diagonalizable, $\dim E_{\lambda i}=$ multiplicity of $\lambda i$. $Ax=\lambda x\implies $ $A^kx={\lambda}^kx$. That's all I know for the question. Could someone give some ...
4
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3answers
112 views

If $A^3=A^2$ then $A^2$ is diagonalizable.

Let $A\in \mathbb{k}^{n\times n} $. Prove that if $A^3=A^2\ne0$ then $A^2$ is diagonalizable. Could you give me any hints on how to prove it?. I can't use the minimal polynomial, since we haven't ...
0
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1answer
34 views

Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement: "Given an object with a ...
0
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1answer
42 views

$A$ has $n$ distinct eigenvalues and $AB=BA$ with $AC=CA$

Suppose $A,B$ and $C$ are three matrices $n\times n$ matrices such that $A$ has $n$ distinct eigenvalues. Suppose $AB=BA$ and $AC=CA$ then prove that $BC=CB$.. Suppose $A$ has $n$ distinct ...