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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Proof of a theorem on simultaneous diagonalization from Hoffman and Kunze.

Now I am reading Linear Algebra from the book of Hoffman and Kunze second edition. I am trying to understand theorem $8$ on pg number $207$ which is based on Simultaneous diagonalization. I have seen plenty of proofs on this simultaneous…
user371231
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Let $S$ be a diagonalizable matrix and $S+5T=I$. Then prove that $T$ is also diagonalizable.

My solution: Since $S$ is diagonalizable, so we can write $S=P^{-1}DP$, where $P$ is an invertible matrix and $D$ is a diagonal matrix. Now $5T=I-S=P^{-1}P-P^{-1}DP=P^{-1}(I-D)P$. So $T=P^{-1}\frac{1}{5}(I-D)P$. Since $I-D$ is also a diagonal…
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$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a field $F$. Let $A$ be a fixed matrix in $V$. Let $T$…
Cousin Dupree
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Over which fields (besides $\mathbb{R}$) is every symmetric matrix potentially diagonalizable?

Over which fields (besides the well-known $\mathbb{R}$) is every symmetric matrix potentially diagonalizable? A matrix is potentially diagonalizable in a field $F$ if it is diagonalizable in the algebraic closure of $F$. It appears to me that the…
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$A$ and $A^2$ have same characteristic polynomial

Is it possible to have a non-identity $2 \times 2$ diagonalizable, invertible, complex matrix $A$ s.t characteristics polynomials of $A$ and $A^2$ are the same? I am not getting any hint even how to create one. I can start with two different…
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Any Nilpotent Matrix is not Diagonalizable

I'm trying to go about the proof that any matrix that is nilpotent (i.e. $\exists N \in\Bbb N. A^N = \mathbf{0}$) cannot be diagonalizable. I believe that the best way to go about this is by showing that a given eigenvalue's geometric multiplicity…
Jennifer
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Suppose $e^A = A$, prove that $A$ is diagonalizable

Suppose $e^A = A$, prove that $A$ is diagonalizable, where A is a matrix. What I have tried to do is write $A= D + N$, where $D$ is diagonalizable, $N$ is nilpotent and $DN = ND$. Since $N$ is nilpotent, there exist a minimal $n$ such that…
user138017
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Proving that a symmetric matrix is positive definite iff all eigenvalues are positive

This has essentially been asked before here but I guess I need 50 reputation to comment. Also, here I have some questions of my own. My Proof outline: (forward direction/Necessary direction): Call the symmetric matrix $A$. Write the quadratic form…
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An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$

I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze. Here is the question: Let $T$ be a a linear operator on an $n$-dimensional space, and suppose that $T$ has $n$ distinct…
Cousin Dupree
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Simultaneous Diagonalizability of Multiple Commuting Matrices

I know that for two given diagonalizable matrices $A_1$ and $A_2$, they commute if and only if they are simultaneously diagonalizable. I was wondering if a similar condition held for multiple pairwise commuting matrices. Specifically, if we have a…
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Show that $B$ is diagonalizable if If $AB=BA$ and $A$ has distinct real eigenvalues

We were asked to prove the following: Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable. It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both…
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When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where \begin{equation} P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \cdot^{\cdot^{\cdot}} & 1 & 0 \\ 0 & \cdot^{\cdot^{\cdot}} &…
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Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is diagonalizable ? I only found out that $A=A^{-1}$ implies…
Peter
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Minimal polynomial of diagonalizable matrix

Prove that a matrix $A$ over $\mathbb{C}$ is diagonalizable if and only if its minimal polynomial's roots are all of algebraic multiplicity one.
Jenni201
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trace , determinant and which of the following are true(NBHM-$2014$)

Let $A \in M_2(\mathbb R)$ be a matrix which is not a diagonal matrix . Which of the following statements are true?? a. If $tr(A)=-1$ and $detA=1$, then $A^3=I$. b. If $A^3=I$, then $tr(A)=-1$ and $det(A)=1$. c. If $A^3=I$, then $A$ is…