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Questions tagged [diagonalization]

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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Theoretical questions about characteristic polynomials, diagonalizability, rank and similarity

I am new to Linear Algebra and would like some feedback regarding my answer to the following question A and B are two square matrices of order n. Prove or refute (with a counterexample) the following ...
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1answer
18 views

Diagonalizability of matrix over C and R respectively

I am new to linear algebra, and was asked the following question: $A$ is a matrix of order n x n, $n \geq 2$, with characteristic polynomial $p(λ)=λ^n-1$. Is $A$ diagonalizable over R? over C? If $A$ ...
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1answer
13 views

Eigenspace for a linear transformation + diagonalizability

I am new to linear algebra, and was asked the question below. I am just looking for some feedback regarding my proposed answer. $T:R_4[x] \to R_4[x]$ is a linear transformation defined by $T(p(x))=p(...
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2answers
13 views

Eigenvalues and eigenvectors in combined Linear Transformations

I am new to linear algebra, and cannot work out the following question, despite the fact that I have been thinking about it for a long while. Let $V$ be a linear space of n dimensions over R, and let ...
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1answer
13 views

Diagonalizability in relation to characteristic polynomial and row equivalence

I am new to linear algebra, and am unsure re the following question: True or False? Let A and B be matrices of n x n. If A and B are diagonalizable and they have the same characteristic polynomial, ...
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0answers
16 views

Similar matrices if only difference is diagonalizable/non-diagonalizable

I am new to linear algebra, and am just looking for some feedback regarding the following solution: True or false? 1.$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$\begin{pmatrix}2&0\\0&...
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2answers
12 views

Diagonalizability in relation to squaring and transposition

True or False? Let A be a square matrix If $A$ is diagonalizable, then $A^2$ is diagonalizable. If $A$ is diagonalizable, then $A^t$ is diagonalizable. Re 1, my answer is that it is correct, but I ...
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3answers
47 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
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1answer
31 views

Diagonalizability over $\mathbb{C}$ and $\mathbb{R}$ respectively

I am new to Linear Algebra, and would love some feedback regarding the following question, which I found a bit confusing: $$A = \begin{Bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&...
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1answer
22 views

Diagonalise a sparse (symmetric) matrix with elements only on some diagonals

Is there an analytical way or a good approximation or any other mathematical method to diagonalise a sparse (symmetric) matrix with elements only onsome diagonals? For example $$ \begin{bmatrix} B &...
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2answers
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Diagonalizing matrix with fractions [on hold]

I'm revising for an exam in linear algebra, and I've found myself stuck on this one specific exercise. I'm supposed to decide a matrix $P$ and a diagonal matrix $D$ from my matrix $H$ (which I'll ...
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1answer
23 views

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$.

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$. Is the statement true? I think the statement is true. My Attempt : I ...
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3answers
48 views

Can a matrix be not a multiple of identity, have repeated eigen values and still be diagonalizable?

The question: Diagonalisability of 2×2 matrices with repeated eigenvalues suggests that if a matrix has all its eigen values distinct, it must be diagonalizable. However, any multiple of the ...
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2answers
33 views

How to find eigenvector when the roots are complex

Let $A=\begin{pmatrix}1&-4\\1&1\end{pmatrix}$ then I want to diagonalize this matrix. Doing it's characteristic polynomial I find out that $\lambda_{1,2}=1\pm2i$. Then it's diagonal matrix $$D=...
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0answers
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prove/disprove if each two in $n$ operators can be diagonalizable simultaneously then all can be diagonalizable simultaneously

I have an idea that for $n$ diagonalizable operators $A_1, A_2, ..., A_n \in \ell(V)$. if each $A_i, A_j$ can be diagonalizable simultaneously then all of them can be diagonalizable simultaneously. ...
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3answers
31 views

$A$ square matrix, $\mathrm{rank}(A)=3,$ characteristic polynomial of A is $x^2(x-1)(x-2) \Rightarrow A$ is diagonalizable

I've been trying to prove or disprove the following statement: Let $A$ be a square matrix such that $\mathrm{rank}(A)=3$. Prove or disprove that if the characteristic polynomial of $A$ is $x^2(x-1)(...
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1answer
29 views

$A,B\in \mathbb R^{n\times n}$ share $n$ common linearly-independent eigenvectors $\Rightarrow AB=BA$.

I've been trying to prove the following statement: Let $A,B\in \mathbb R^{n\times n}$ be square matrices such that they share $n$ common linearly-independent eigenvectors. Then $AB=BA$. Everything ...
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1answer
16 views

Diagonal Matrix Problem

Could someone check if the solution of the problem is right? Problem: Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix ...
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1answer
42 views

Analytic functions and diagonalisation of matrices.

If I have an analytic function $f$ of a square matrix A (like sin(A)), then I know that if the matrix diagnosable then it is possible to find a matrix $$D = P^{-1}AP \tag{1}$$. Then for a function $f(...
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1answer
56 views

Unitary Matrices Proof

Problem: Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix $U \in \mathbb{C}^{n\times n}$ such that $U^*$A$U$ and $U^*$$...
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2answers
49 views

Proof of Jordan-Chevally-Decomposition

Let A be a square matrix over $\mathbb{C}$, prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is Nilpotent and $DN = ND$. I can see that any ...
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2answers
33 views

For a square matrix A over $\mathbb{C}$, Proofs that matrices D and N exist with A=D+N under different conditions

(i) D is Diagonalizable This one i believe to be fairly straightforward, if D is diagonalizable then we can allow $D^t = I$ (where I is the identity) and therefore D id diagonalizable and therefore A=...
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1answer
42 views

Diagonalization to power of matrix

$\left(\array{ 2&3 \\ 5& 1 }\right)^{20}$ $(\lambda-1)(\lambda-2)-15=0$ $\lambda = (3\pm \sqrt{61})/2$ Is this problem should be solved in this method? Lambda is not tidy.
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1answer
49 views

Power of matrix using diagonalization

First one $$\begin{pmatrix} 2& 3\\5 & 1 \end{pmatrix}^{20}$$ Second one $$A=\begin{pmatrix} 4&0& 0\\0 & 3&0\\2 &0&2 \end{pmatrix}^{20}$$ $$P=\begin{pmatrix} 1&0&...
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2answers
27 views

How to rewrite matrix formula for Diagonalizable matrix $A=PDP^{-1}$

I am working on an old exam containing a question about Diagonalizable matrix, I am quite confident about the subject overall but there is one simple thing that bothers me, a lot! We are given the ...
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1answer
48 views

Proof that spectrum of a matrix is subset of the positive real numbers

So my given problem is: $Let\,\, A \in \mathbb{C}^{n \,\times \,n}\,\, be\,\, such\,\, that\,\,\\ \forall x\in \mathbb{C}^n : \langle\,Ax,x\rangle \geq 0 \\ where \,\, \langle\,\cdot,\cdot\rangle \,\,...
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1answer
49 views

Show that a matrix is diagonal [duplicate]

My task given by my professor was the following: $Let \,\, A, \,\, B \in \mathbb{C}^{n\times n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that ...
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1answer
27 views

Proof on unitary diagonalisable matrices

My task is the following: $Let\,\, A,U \in \mathbb{C}^{n\times n} \,\,and\,\,assume \,\,that \,\, U \,\,is \,\,unitary \,\, and \,\, that \,\,U^*AU \,\,is\,\, a\,\, diagonal\,\, matrix.\\Show \,\, ...
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2answers
38 views

Eigenvalues of diagonal matrix

Problem: Let $A$ ∈ $\Bbb C^{n×n}$ and let $A$ be a diagonal Matrix with entries $\lambda_1, \ldots, \lambda_n$ ∈ $\mathbb{C}$ Determine spec($A$*$A$) I think it is clear that the spec ($A$ *$A$) ...
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0answers
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Symmetric matrix with n rows has n linearly independent eigenvectors?

I'm trying to better understand SPD matrices and have started with symmetric matrices. It's apparent to me that symmetric matrices have real eigenvalues, but how to show that the algebraic ...
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3answers
73 views

Let $A= \begin{pmatrix} 8&2 \\ -8&-2 \end{pmatrix}$. Find the entry in the first row and second column of $A^{2014}$

I have tried diagonalizing the matrix and obtained: $A=PDP^{-1}$. Where: $P=\begin{pmatrix} 1&1 \\ -4&-1 \end{pmatrix}$ $D=\begin{pmatrix} 0&0 \\ 0&6 \end{pmatrix}$ $P^{-1}=\...
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3answers
47 views

Example of diagonalisable matrix with given property

Let $A\in M_5(\mathbb C)$ satisfying $(A^2-I)^2=0$ and $A$ is not diagonal matrix. Then I have To find matrix A But I tried but adding some terms in up to diagonal Nilpotency occur Which prevent ...
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0answers
30 views

Spectral decomposition of an operator

Given the operator: $$\begin{pmatrix} i & 0 & - 4\\ 0 & - 3i & 0\\ 2 & 0 & - i \end{pmatrix}$$ Now $det(\lambda1-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$, I know a theorem that says ...
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1answer
57 views

Find a matrix $P$ that diagonalizes the matrix $A$, and determine $P^{-1}AP$

$A =\begin{bmatrix} -14 & 12 \\ -20 & 17 \\ \end{bmatrix} $ I did this by first calculating the eigenvalues which turn out to be $λ = 2$ and $λ = 1$ Then I calculated the eigenvectors by ...
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0answers
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If A is anti-hermitian, show that $|\det(1+A)|^2 \geqslant 1$

Given that A is anti-hermitian, i.e. $A^\dagger = -A$, show, by diagonalising $iA$, that $$|\det(1+A)|^2 \geqslant 1$$ $\hspace{3mm}$ So what I thought was that I need to show that $\det(1+A)[\det(1+...
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1answer
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Simultaneous Diagonalisation [duplicate]

Im stuck at this problem Find an invertible Real Matrix $P$ such that $P^{-1}AP$ and $P^{-1}BP$ are both diagonal where $A$ and $B$ are real matrices. a) $A=\begin{bmatrix} 1&2\\ 0&2\\ \...
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3answers
43 views

True or False, diagonalization problem

Let $B_c= \left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}$, $T:\mathbb{R}^4\rightarrow \mathbb{R}^4$ a linear operator such that \begin{equation} \det\left[T-I\lambda\right]_{B_{c}}=(2-\lambda)^...
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0answers
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Compare classical and modified algorithm of Cholesky

Let $\textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $\textbf{A} = \textbf{R}^T\textbf{D}\textbf{R}$ and the classical factorization $\textbf{A} = \textbf{R}_c^T\textbf{R}...
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1answer
31 views

Diagonalizable matricies and eigenvalues

Let $A$,$B$,$C$ be three different real $3 \times 3$ matricies with the following properties: $A$ has the complex eigenvalue $\lambda=3-5i$ $B$ has eigenvalues $\lambda=0$, $\lambda=5$, $\...
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2answers
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Faster way to calculate the inverse of matrice $C $(when diagonisable.. $C-1AC = D$)

We found the orthonormal basis for the eigen spaces. We got $C$ to be the matrix ...
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2answers
45 views

Prove diagonalizability of operator $T$ [closed]

I got homework to prove some question and after almost 5 hours I gave up. The questions are: 1) operator $T : \Bbb R^n\to \Bbb R^n$, prove that $\operatorname{Im}(T)∩ \operatorname{Ker}(T)={0}.$ 2) ...
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2answers
35 views

Find invertible matrix $Q$

From the book "Bilinear forms and their matrices" by Prof. Joel Kamnitzer: Consider: $$ A=\pmatrix{0 & 4 \\ 4 & 2} $$ After doing simultaneous row and column operations we reach: $$ Q^...
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2answers
45 views

Simultaneous diagonalisation of quadratic forms

Is there a linear transformation that simultaneously reduces the pair of real quadratic forms $$x^2-y^2$$ and $$2xy$$ to diagonal forms? My attempt I know that neither of these forms are positive ...
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0answers
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Prove that all normal matrices are semi-simple using Schur's decomposition

Is there any elegant proof that shows that all normal matrices are semi-simple that comes from Schur's decomposition or its corrolaries? There is a proof that normal matrices are unitary diagonizable ...
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2answers
32 views

Determine all real numbers that can be eigenvalues of operator A.

Problem: Let $A$ be some linear operator such that: $$ (A^{2006}-I)^{2006}-I=0. $$ Determine all real numbers that can be eigenvalues of operator $A$. Question: How to solve this problem? My ...
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22 views

Diagonalize matrix and find rotation-dilation matrix

Diagonalize the following matrix: \begin{bmatrix}0&1\\-5&-2\end{bmatrix} Also write it in form $PCP^{-1}$, where $C$ is a rotation-dilation matrix. How would I approach this problem? How do ...
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0answers
25 views

Weakly Diagonally Dominant with Positive Diagonals

Suppose $A \in \mathbb{R}^{n\times n}$ (not necessarily symmetric) is weakly diagonally dominant with positive diagonals. Is it true that all eigenvalues of $A$ are non-negative (in the case of ...
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3answers
47 views

Does a $2\times2$ matrix B exist such that B is diagonalizable but B is not invertible?

Is it possible? I know that for a matrix to be diagonalizable it needs sufficient linearly independent eigenvectors to make up invertible matrix P, but does that mean B itself must also be invertible?
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1answer
32 views

Basis of a transformation matrix for diagonal matrix

So I have this question here which says: Let $T:\Bbb R^3\to\Bbb R^3$ defined by $$T\left(\begin{matrix}x_1 \\ x_2 \\ x_3 \end{matrix}\right)=\left(\begin{matrix}-x_1+7x_2-x_3 \\ x_2 \\ 15x_2-...
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1answer
55 views

When is this matrix diagonalizable?

Given the matrix $$A = \left[ \begin{matrix} 2 & 2 & h & 6 \\ 0 & 4 & 2 & -2 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 4 \\ \end{matrix} \...