# 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.

1,444 questions
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### How come a matrix not diagonalizable if the geometric multiplicity is less than the algebraic multiplicity?

I understand that the algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. I also understand that the geometric multiplicity is the dimension ...
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### If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
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### Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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### Properties of $n \times n$ complex matrix with $A^m = I$

If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$? The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since ...
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### Alternative definition of diagonalisable transformation

Supposedly a transformation $T: V \to V$ is diagonalisable iff there exists a basis of $V$ consisting only of eigenvectors of $T$. Can someone show me why this is true? I don't really know where to ...
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### Eigenvalues, diagonalization and convergence of matrices

I am trying to wrap my head around some basic results in Linear Algebra. I am trying to avoid more abstract concepts like Rank-Nullity, and stay in simple properties at an introductory level. (I've ...
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### If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable? [closed]

If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable?
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### show that $C(C[u])=K[u]$

Let $E$ be a $K$-vector space of dimension $n$, and $u$ be a diagonalizable linear map from $E$ to $E$. Let $C[u]$ be the set of linear maps from $E$ to $E$ which commute with $u$. In other words,...
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### What would be the restrictions to diagonalization of this type transformation of Laplacian matrix?

This question is a little specific. I have read about graph theory and saw in one place the following transformation of an Laplacian matrix $L$: \begin{eqnarray} ULW=\begin{bmatrix} -(l_{11}-l_{22})&...
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### Find 3x3 matrix by determinant and 2 eigenvalues/-vectors

I have two eigenvectors: $(2, 1, -1)'$ with eigenvalue $1$, and $(0, 1, 1)'$ with eigenvalue $2$. The corresponding determinant is $8$. How can I calculate the $3\times3$ symmetric matrix $A$ and $AP$?...
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### Can you tell how many eigenvectors a matrix has from just the characteristic equation?

If the equation has a repeated root, can you tell without evaluating in the matrix if that repeated root corresponds to more than one eigenvector?
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### Is $A \in \mathcal{M}_n(\mathbb{C})$ diagonalizable?

$A \in \mathcal{M}_n(\mathbb{C})$ such that $A^2$ has got $n$ distinct non zero eigenvalues. Show that A is diagonalizable. Attempt : As $A^2$ has got $n$ distinct non zero eigenvalues. The ...
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### Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...
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### Routines for diagonalization of banded hermitian matrices?

I have a problem in which I need to diagonalize matrices with many thousands of complex elements three times. I know that the matrices are hermitian and sparse. Specifically, they consist of 9 bands ...
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### Similarity between a matrix and diagonal matrix

How to prove a matrix and a diagonal matrix are similar? Is there some rule to follow or are there some steps to follow? Like: suppose a matrix $A$ = something, prove that $A$ is similar to a ...
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### What can I infer about the eigenvalues of the sum of two matrices with known eigendecompositons

I have two matrices and I know the eigenvalues and eigenvectors of both, it also happens to be the case that they share the same set of eigenvalues. I now need to sum the two matrices and take the ...
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### 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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### 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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### Diagonalizing matrix with fractions [closed]

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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### 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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### 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 ...
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 $$D=... 0answers 21 views ### prove/disprove if each two in n operators can be diagonalizable simultaneously then all can be diagonalizable simultaneously I have an idea that for n diagonalizable operators A_1, A_2, ..., A_n \in \ell(V). if each A_i, A_j can be diagonalizable simultaneously then all of them can be diagonalizable simultaneously. ... 3answers 35 views ### A square matrix, \mathrm{rank}(A)=3, characteristic polynomial of A is x^2(x-1)(x-2) \Rightarrow A is diagonalizable I've been trying to prove or disprove the following statement: Let A be a square matrix such that \mathrm{rank}(A)=3. Prove or disprove that if the characteristic polynomial of A is x^2(x-1)(... 1answer 41 views ### A,B\in \mathbb R^{n\times n} share n common linearly-independent eigenvectors \Rightarrow AB=BA. I've been trying to prove the following statement: Let A,B\in \mathbb R^{n\times n} be square matrices such that they share n common linearly-independent eigenvectors. Then AB=BA. Everything ... 1answer 19 views ### Diagonal Matrix Problem Could someone check if the solution of the problem is right? Problem: Let A, B \in \mathbb{C}^{n\times n} be selfadjoint ,such that [A,B] := AB − BA = 0 Show that there is a unitary matrix ... 1answer 49 views ### Analytic functions and diagonalisation of matrices. If I have an analytic function f of a square matrix A (like sin(A)), then I know that if the matrix diagnosable then it is possible to find a matrix$$D = P^{-1}AP \tag{1}$$. Then for a function f(... 1answer 57 views ### Unitary Matrices Proof Problem: Let A, B \in \mathbb{C}^{n\times n} be selfadjoint ,such that [A,B] := AB − BA = 0 Show that there is a unitary matrix U \in \mathbb{C}^{n\times n} such that U^*AU and U^*$$...
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 nilpotent ...