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.

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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 ...
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1answer
18 views

Opposite determinant in Autonne-Takagi factorization

Let us consider a complex symmetric matrix in $M_2(\mathbb C)$ \begin{equation} A = \begin{pmatrix} x_1+ix_2 & x_3 \\ x_3 & -x_1+ix_2 \end{pmatrix} \end{equation} where the $x_i\in \mathbb R,\;...
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1answer
34 views

Bring linear map to a new basis

I have a linear map: $$A = \begin{pmatrix} -1 & 3 & -1 \\ -3 & 5 & -1\\ -3 & 3 & 1 \end{pmatrix}$$ Find the basis and its matrix upon transition to which the ...
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3answers
45 views

Proving the following matrix is diagonalizable

I'm asked to prove that the matrix $A\in M_{n}(\mathbb C)$ that satisfy $A^8+A^2=I$ is diagonalizable. I've tried looking at the equation $x^8+x^2-1=0$ and determining whether $M_A$ has any repeating ...
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Diagonalizable matrix over R

I had study about solving linear difference equation system by using results of linear algebra. But I have a problem with the following exercise: Suppose that $A$ is a matrix with real entries, $A$ ...
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2answers
17 views

For general 2d complex traceless $\sigma$, the U that diagonalises $\sigma^\dagger \sigma$ leaves $U^\dagger \sigma U$ with zeros on the diagonal?

The general $2\times2$ complex traceless matrix can be written in terms of the Pauli matrices $\sigma = a^i \sigma^i$. Consider also its conjugate $\sigma^\dagger=a^{*i}\sigma^i$. I was interested in ...
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1answer
30 views

What is wrong with this bijection from all naturals to reals between 0 and 1?

I will use binary. I claim to have a bijection $f \colon \mathcal{N} \to \left[ 0, 1 \right)$ where $\mathcal{N}$ is the set of natural numbers $\left\{ 0, 1, 10, 11, 100, \dotsc \right\}$ as follows:...
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Calculate power of a matrix using jordan form

I need to calculate: $$ \begin{bmatrix} 1&1\\ -1&3 \end{bmatrix}^{50} $$ The solution i have uses jordan form and get to: There are some points that i dont understand: $1.$ In the right ...
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41 views

Diagonal matrix congruent to a symmetric complex matrix

Given the matrix: $$A=\begin{pmatrix}i&1\\1&-i\end{pmatrix}$$ Find a matrix $P$ such that $P^T A P$ is diagonal, how should I go about this? we know from Sylvester's theorem that $A$ is ...
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Diagonalizablity and complex conjugates linear algebra proof problem

Let $A \in M_{n \times n}(\Bbb R)$. Let $T_{\Bbb R}:\Bbb R^n \to \Bbb R^n$ and $T_{\Bbb C}:\Bbb C^n \to \Bbb C^n$ be the corresponding linear mapps(defined by $T_{\Bbb R}(x)=Ax$ and $T_{\Bbb R}(v)=Av$ ...
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30 views

Diagonalisability of matrices

There are $2 \times 2$ matrices $A$, $B$, $C$, and $D$ with following properties. $A$ is symmetric and negative definite. $B$ is orthogonal and $\det(B)=1$. $C$ is orthogonal and $\det(C)=-1$. $D$ is ...
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29 views

Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A=PDP^{-1}$. [closed]

Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A=PDP^{−1}$. Use your answer to find an expression for A$^6$ in terms of $P$, a power of $D$, and $P−1$ in that order. link to ...
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Finding diagonal matrix D and invertible matrix C such that $A=C^{-1}DC$

Here is the full question: Consider the matrix: \begin{equation*} A= \begin{pmatrix} -3 & 5 \\ -2 & 4 \end{pmatrix} \end{equation*} Find a diagonal matrix $D$ and an invertible matrix $C$ ...
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1answer
21 views

diagonal form of a complex matrix

Let us consider a complex symmetric matrix \begin{equation} A = \begin{pmatrix} a+ib & c \\ c & -a+ib \end{pmatrix} \end{equation} where the coefficients $a,b,c$ are real. I am interested in ...
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1answer
32 views

Is this givens can define a matrix?

(a) Let $A\in M_{5\times 5}(\mathbb{C})$ s.t: $$dim~N(A) =3$$ $$dim~N(A-I)=2$$ $$p_A(x)=x^2(x-1)^3$$ (b) Let $A\in M_{5\times 5}(\mathbb{C})$ s.t $$dim~N(A) =3$$ $$dim~N(A-I)=0$$ $$p_A(x)=x^4(x-1)$$ ...
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If $A$ is an $n \times n$ matrix such that $A^5 = 0$, which of the following is true?

I have this question from my textbook: Given $A$ an $n \times n$ matrix such that $A^5 = 0$, which of the following is true? $(a)$ $A$ is invertible. $(b)$ $A = 0$. $(c)$ $A$ is diagonalizable. $(...
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1answer
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Singular value decomposition (SVD) for non-symmetric square real matrix contradicts spectral theorem?

From Bretscher's Linear Algebra with Applications: where $A$ is a real matrix in $ \mathbb{R}^{n \times m}$ and the singular values of $A$ are the square roots of the eigenvalues of the symmetric $A^...
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Why is my matrix not unitary?

I've to find a matrix unitary matrix U, so that $U^* A U= diag(A)$. Or in other words: I've to find a unitary Matrix U that diagonalizes A. $A = \begin{bmatrix}i&1&i&-1\\-1&i&1&...
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Proof verification: a basic proof related to diagnonal block matrix. (formal verification)

So, i want to prove the following statement: let A be diagonal block matrix with the blocks $ A_{1},...A_{r}$ then $xI-A=\left(\begin{array}{cccc} xI-A_{1}\\ & xI-A_{2}\\ & & \ddots\\ ...
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1answer
31 views

Can I use diagonalization to find the $k$'th power of a matrix if $k < 1$?

I know that we can use the diagonalization of a matrix: $$\hspace{9.4cm} A = V \Lambda V ^{-1} \hspace{9cm} (1)$$ (where $V$ is the matrix of eigenvectors and $\Lambda$ is the (diagonal) matrix of ...
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prove that if $ T^2=T $ then T is diagonalizable operator ( over finite dimension vector space)

I know its a very basic question in linear algebra, and I actually solved it myself. But something's tells me there's a lot more ways to prove it, and I'd like to see some other informative ways that ...
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1answer
87 views

If $B$ is a diagonalizable $n\times n$ matrix, prove that $B^{2000}$ is also diagonalizable.

Would it be correct to prove this by referencing a theorem in our text that says that if $x$ is an eigenvector of $B$ corresponding to the eigenvalue t ,then x is an eigenvector of $B^m$ corresponding ...
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26 views

Normalization of Generators in SU(N)

I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
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1answer
30 views

Which is the rotated frame in which the given matrix transforms into this block matrix? [UPDATED]

Given a matrix $A=\begin{pmatrix} a & b & c \\ b & a & -c \\ c & -c & d \end{pmatrix}$ with positive $a,b,c,d \in R$ , which are the angle $\theta$ and rotation axis that ...
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35 views

Any complex $2\times 2$ matrix is of one of the two forms: proof.

Let $A$ be a $2\times 2$ complex matrix,then show that $A$ is similar over $\mathbb C$ to either $\begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix}$ or $\begin{bmatrix} a & ...
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1answer
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 ...
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1answer
25 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 $...
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1answer
23 views

Prove: block matirx {{A,-A},{-A,A}} is diagonalizable for diagonalizable matrix A

Given diagonalizable $n \times n$ matrix $A$ ($A = PDP^{-1}$, where D is diagonal matrix). How can I prove that $$ \left[\begin{matrix} A, & -A \\ -A, & A \\ \end{matrix} \right] $$ is ...
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38 views

Diagonalization of a block matrix

I've just started learning about eigenvectors, eigenvalues and similar matrices, so I apologize if this question is simple. I have a nxn matrix M, which is diagonalizable. I have to show that the ...
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Can someone please explain what does $ D= P^{-1} AP$ mean in simple terms?

I know how determine whether or not a matrix is diagonalizable, but after that the question will usually ask me to find an invertible matrix P and a diagonal matrix D such that $$ D= P^{-1} AP$$. I ...
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1answer
82 views

If $A$ is compact and self-adjoint, then $A=U^\ast DU$ for some orthogonal $U$ and diagonal $D$

Let $H$ be a $\mathbb R$-Hilbert space $A\in\mathfrak L(H)$ be compact and self-adjoint $J:=\mathbb N\cap[0,|\sigma(A)\setminus\{0\}|]$ and $(\mu_j)_{j\in J}$ be an enumeration of $\sigma(A)\setminus\...
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1answer
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 $A .$ $(b)$ ...
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1answer
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 ...
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1answer
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 ...
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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 $A$ ...
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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 ...
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1answer
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 ...
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1answer
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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 $A$ ...
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Not diagonalizable matrix

I have the following matrix of dimension $n\times n$: $A = \begin{pmatrix} a_{1} & 1 & \\ & a_1 & 1 & \\ & & \ddots & 1\\ & &&...
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Confusion about diagonalization of a 2-by-2 matrix

I have matrix $$ M = \begin{bmatrix}0&r\\k^{-1}&0\end{bmatrix} $$ where $k,r$ are real constants.The book says that a matrix $P$ can diagonalize $M$ such that $M=P^{-1}DP$ where $D=$ diagonal ...
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1answer
29 views

Orthogonally Diagonalize the matricies

The Question given was "Orthogonally diagonalize the matrices, giving an orthogonal matrix P and diagonal matrix D." I was given eigenvalues -3 and 15 and Matrix A= $$ \begin{bmatrix} 5 & 8 & ...
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An orthogonal matrix in $\mathbb{R}^{3\times3}$ with real eigenvalues is diagonalizable [duplicate]

I know there are two non trivial (i.e. if we solve these two cases the other cases are trivial) cases: $\lambda_{1,2,3}=1$ and: $\lambda_1=1,\lambda_{1,2}=-1$ I have been trying to use generalized ...
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Projections on eigenspaces of diagonalizable matrix

Let $A \in \mathbb{C}^{n \times n}$ be a diagonalizable matrix and let $\mathfrak{A_1}, \dots \mathfrak{A}_n$ the eigenspaces such that $$\mathbb{C}^{n} = \mathfrak{A_1}\oplus \dots \oplus\mathfrak{...
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1answer
26 views

Prove that a matrix $\mathbf{A} \in M_n$ is similar to a Hermitian matrix if and only if it is diagonalizable and has real eigenvalues

So if the matrix $\mathbf{B}$ is Hermitian, that also means its diagonalizable. And if $\mathbf{A}$ is similar to $\mathbf{B}$ then there exists an invertible matrix $\mathbf{S}$ such that $\mathbf{A}...
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1answer
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 ...
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1answer
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-...
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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 $k > 0$ , ...
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constraint optimization solver

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)...
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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 $n$ ...
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1answer
22 views

Diagonalization of an Hermitian matrix

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 ...

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