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.

A square matrix $A$ is diagonalisable if there is an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. One can view $P$ as a change of basis matrix so that, if $A$ is viewed as the standard matrix of a linear map $T$ from a vector space to itself in some basis, it is equivalent to say there exists an ordered basis such that the standard matrix of $T$ is diagonal. Diagonal matrices present the eigenvalues of the corresponding linear transformation along its diagonal. A square matrix that is not diagonalizable is called defective.

Not every matrix is diagonalisable over $\mathbb{R}$ (i.e. only allowing real matrices $P$). For example, $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

Diagonalization can be used to compute the powers of a matrix $A$ efficiently, provided the matrix is diagonalizable.

Diagonalization Procedure :

Let $A$ be the $n×n$ matrix that you want to diagonalize (if possible).

  • Find the characteristic polynomial $p(t)$ of $A$.
  • Find eigenvalues $λ$ of the matrix $A$ and their algebraic multiplicities from the characteristic polynomial $p(t)$.
  • For each eigenvalue $λ$ of $A$, find a basis of the eigenspace $E_λ$. If there is an eigenvalue $λ$ such that the geometric multiplicity of $λ$, $dim(E_λ)$, is less than the algebraic multiplicity of $λ$, then the matrix $A$ is not diagonalizable. If not, $A$ is diagonalizable, and proceed to the next step.

  • If we combine all basis vectors for all eigenspaces, we obtained $n$ linearly independent eigenvectors $v_1,v_2,…,v_n$.

  • Define the nonsingular matrix $$P=[v_1\quad v_2\quad …\quad v_n]$$
  • Define the diagonal matrix $D$, whose $(i,i)$-entry is the eigenvalue $λ$ such that the $i^{th}$ column vector $v_i$ is in the eigenspace $E_λ$.
  • Then the matrix A is diagonalized as $$P^{−1}AP=D$$

References:

Diagonal Matrix on Wikipedia

Matrix Diagonalization on Wolfram MathWorld

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Square root of Positive Definite Matrix

Let $A$ be an $n\times n$ positive definite matrix. Show that there exists a unique positive definite matrix $B$ such that $B^2=A$. I do know the existence. But what about the uniqueness? Would you help me out? Thank you.
XLDD
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Diagonalization of a projection

If I have a projection $T$ on a finite dimensional vector space $V$, how do I show that $T$ is diagonalizable?
smanoos
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Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have some group $G$ of matrices, where $G$ is one…
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Prove that Hermitian matrices are diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are orthogonal. If $A=A^H;\;\;\lambda_1,\lambda_2$ be…
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What's so useful about diagonalizing a matrix?

I'm told the the purpose of diagonalisation is to bring the matrix in a 'nice' form that allows one to quickly compute with it. However in writing the matrix in this nice diagonal form you have to express it w.r.t. a new eigenvector basis. But…
user2520938
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If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
Jane
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Eigenvalues of outer product matrix of two N-dimensional vectors

I have a vector $\textbf{a}=(a_1, a_2, ....)$, and the outer product $M_{ij}=a_i a_j$. What are the eigenvalues of this matrix? and what can you say about the co-ordinate system in which $M$ is diagonal? I have proved that the only eigenvalue of the…
user23238
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Are these square matrices always diagonalisable?

When trying to solve a physics problem on decoupling a system of ODEs, I found myself needing to address the following problem: Let $A_n\in M_n(\mathbb R)$ be the matrix with all $1$s above its main diagonal, all $-1$s below its diagonal, and $0$s…
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Find large power of a non-diagonalisable matrix

If $A = \begin{bmatrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then find $A^{30}$. The problem here is that it has only two eigenvectors, $\begin{bmatrix}0\\1\\1\end{bmatrix}$ corresponding to eigenvalue $1$ and…
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Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with it for the first time at a note about a simple…
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Show that if $A^{n}=I$ then $A$ is diagonalizable.

Suppose $A$ is an $m \times m$ matrix which satisfies $A^{n}=1$ for some $n$, then why is $A$ necessarily diagonalizable. Not sure if this is helpful, but here's my thinking so far: We know that $A$ satisfies…
goatman2743
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Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these. It seems to be a well known result that when you take the eigenvectors of $A$ as a basis and diagonalize $B$ with it then you get a block…
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Which matrices $A\in\text{Mat}_{n\times n}(\mathbb{K})$ are orthogonally diagonalizable over $\mathbb{K}$?

Update 1. I still need help with Question 1, Question 2' (as well as the bonus question under Question 2'), and Question 3'. Update 2. I believe that all questions have been answered if $\mathbb{K}$ is of characteristic not equal to $2$. The only…
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Diagonalizable random matrix

Let $p_n$ the probability that a random matrix $M\in\mathcal{M}_n(\mathbb{R})$ such that its entries $(m_{i,j})_{1\leqslant i,j\leqslant n}$ are independant and following an uniform distribution over $[-1,1]$, is diagonalizable. I was wondering how…
Tuvasbien
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Diagonalization: Can you spot a trick to avoid tedious computation?

I am studying for my graduate qualifying exam and unfortunately for me I have spent the last two years studying commutative algebra and algebraic geometry, and the qualifying exam is entirely 'fundamental / core' material - tricky multivariable…
Prince M
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