If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable

Diagonalization Problems and Solutions in Linear Algebra

Problem 84

Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.

Show that the matrix $A$ is diagonalizable.
LoadingAdd to solve later

Sponsored Links


Contents

Hint.

Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is diagonalizable.

Proof.

Since the matrix $A$ satisfies the equation $x^k-1$, the minimal polynomial of $A$ divides $x^k-1$.
Since
\[x^k-1=\prod_{j=0}^{k-1}(x-e^{2\pi i j/k}),\] the roots of $x^k-1$ are all distinct.

Hence the roots of the minimal polynomial are also distinct.
Therefore the matrix $A$ is diagonalizable.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Finite Order Matrix and its TraceFinite Order Matrix and its Trace Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that (a) $|\tr(A)|\leq n$. (b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$. (c) $\tr(A)=n$ if and only if $A=I_n$. Proof. (a) […]
  • Extension Degree of Maximal Real Subfield of Cyclotomic FieldExtension Degree of Maximal Real Subfield of Cyclotomic Field Let $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$. The subfield $\Q(\zeta+\zeta^{-1})$ is called maximal real subfield.   Proof. […]
  • Trace of the Inverse Matrix of a Finite Order MatrixTrace of the Inverse Matrix of a Finite Order Matrix Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix. Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that […]
  • Diagonalizable Matrix with Eigenvalue 1, -1Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]
  • How to Find Eigenvalues of a Specific Matrix.How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]
  • A Matrix Similar to a Diagonalizable Matrix is Also DiagonalizableA Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
  • Degree of an Irreducible Factor of a Composition of PolynomialsDegree of an Irreducible Factor of a Composition of Polynomials Let $f(x)$ be an irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be any polynomial in $F[x]$. Show that the degree of each irreducible factor of the composite polynomial $f(g(x))$ is divisible by $n$.   Hint. Use the following fact. Let $h(x)$ is an […]
  • Diagonalizable by an Orthogonal Matrix Implies a Symmetric MatrixDiagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]

You may also like...

1 Response

  1. 08/27/2016

    […] Remark that if $A$ is a square matrix over $C$ with $A^k=I$, then $A$ is diagonalizable. For a proof of this fact, see If a power of a matrix is the identity, then the matrix is diagonalizable […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Problems and solutions in Linear Algebra
Isomorphism of the Endomorphism and the Tensor Product of a Vector Space

Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space...

Close