If $A^4=I$ then $A$ must be diagonalizable?

Question

Suppose we have a real matrix $A$ which satisfies $A^4=I$, can we determine if $A$ is diagonalizable?

I believe the answer is that we can't because all we know about the matrix $A$ is that it is invertible (otherwise $A^4$ couldn't be an invertible matrix)..

How can I prove it? How can I find such a matrix $A$ which isn't diagonalizable but $A^4 = I$?

The only matrix $A$ I was able to find which satisfies $A^4=I$ is the identity matrix itself but the identity matrix is diagonalizable.

Why? For example : $$A= \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$$ is invertible but $A^4 != I$ — SyndicatorBBB, Dec 26 '14 at 12:26
May bad... sorry, your are right. Anyway another example is given by $A=\begin{pmatrix}0 & 1 \ 1 & 0 \end{pmatrix}$. — Surb, Dec 26 '14 at 12:27
Yes but the matrix : $$A= \left( {\matrix{ 0 & 1 \cr 1 & 0 \cr } } \right)$$ is diagonalizable. — SyndicatorBBB, Dec 26 '14 at 12:29
What is the considered field ? $\mathbb{C}$ ? $\mathbb{R}$ ? Some algebrically closed field ? — Traklon, Dec 26 '14 at 12:33
See this recent thread to see what changes, if we use complex numbers instead, and for a generalization. — Jyrki Lahtonen, Dec 26 '14 at 12:50

user2345215 · Answer 1 · 2014-12-26T12:49:28.237

10

No, consider the rotation by $\pi/2$ in $\mathbb R^2$: $$A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$ Which must satisfy $A^4=I$, but has no (real!) eigenvectors, so it's not diagonalizable.

edited Dec 26 '14 at 12:49

answered Dec 26 '14 at 12:35

user2345215

16,422

Great, exactly what I was looking for! One more thing, Is it possible that $A$ isn't diagonalizable at all? Even under $C$? – SyndicatorBBB Dec 26 '14 at 12:42
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No, since in $\mathbb{C}$ the minimal polynomial of $A$ will have simple roots. See the answers below. – Clement C. Dec 26 '14 at 12:45
Thank you @ClementC. ! Taking all the answers together I finally understand it. – SyndicatorBBB Dec 26 '14 at 12:47
Take a look at this. By the way, there is a matrix that is not diagonalizable in any field. The prime example is given in that link. – Scream Dec 26 '14 at 13:59

WLOG · Accepted Answer · 2014-12-26T13:59:11.957

8

I suppose we are considering $\mathbb{R}$ or $\mathbb{C}$.

The minimal polynomial of $A$ divides $$x^4 -1 = (x-1) (x+1) (x^2 +1) $$

A matrix is diagonalizable over the field $F$ if and only if its minimal polynomial is a product of distinct linear factors over $F$, moreover the minimal and characteristic polynomial have the same roots.

Call $p(x)$ the minimal polynomial of $A$ in the field we are considering.

Thus if $F = \mathbb{C}$ the matrix is always diagonalizable, because the roots of $x^4 -1$ are all differents and so $p(x)$ is a product of distinct linear factors over $\mathbb{C}$ because $p(x) \mid x^4 -1$.

If $F = \mathbb{R}$ we need further information to decide if $A$ is diagonalizable or not.

edited Dec 26 '14 at 13:59

answered Dec 26 '14 at 12:31

WLOG

11,436

I think I understand but I'm a little bit confused... It is not possible for $A$ to be diagonalizable under $R$? – SyndicatorBBB Dec 26 '14 at 12:37
@SyndicatorBBB: If $A$ is $4 \times 4$ no, because in this case the characteristic polynomial has roots $\pm i$, and so also the minimal polyomial – WLOG Dec 26 '14 at 12:41
@SyndicatorBBB: but $\pm i$ are not in $\mathbb{R}$ and so the minimal polyomial isn't the product of linear factors over $\mathbb{R}$ – WLOG Dec 26 '14 at 12:42
If it is diagonalizable, the diagonal entries will be the eigenvalues. But here (if the dimension is at least 4), some of the eigenvalues are complex numbers, not reals: so A cannot be diagonalized in $\mathbb{R}$. However, it is diagonalizable in $\mathbb{C}$. – Clement C. Dec 26 '14 at 12:42
Oh now I understand what you meant. Is it possible that $A$ isn't diagonalizable at all? – SyndicatorBBB Dec 26 '14 at 12:44
1

@SyndicatorBBB: Over $\mathbb{C}$ is always diagonalizable because the roots of $x^4 -1$ are all differents, and so over $\mathbb{C}$ the minimal polynomial is a product of distinct linear factors – WLOG Dec 26 '14 at 12:46
1

@SyndicatorBBB No: in $\mathbb{C}$, $x^4-1=(x-1)(x+1)(x-i)(x+i)$, and the minimal polynomial $\mu_A$ of $A$ divides this polynomial; thus, it has simple roots. And thus $A$ is diagonalizable in $\mathbb{C}$. – Clement C. Dec 26 '14 at 12:47
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Why does the characteristic polynomial of $A$ divide $x^4 -1$ ? – Georges Elencwajg Dec 26 '14 at 12:52
Yes, exactly what I was going to ask.. How did you come up to this conclusion? – SyndicatorBBB Dec 26 '14 at 12:57
He means that the minimal polynomial divides $x^4-1$, and since this has no repeated roots, neither does the minimal polynomial. This is because if $A$ satisfies two polynomial relations, it satisfies their GCD. – Aaron Dec 26 '14 at 13:27
@Aaron Why the minimal polynomial divides $x^4-1$? – SyndicatorBBB Dec 26 '14 at 13:28
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@SyndicatorBBB This comes from the definition (and proof of existence) of minimal polynomial. We know our matrix satisfies that particular polynomial relation. If the minimal polynomial didn't divide it, then either the minimal polynomial would have degree at least five (and wouldn't be minimal degree) or else we could divide $x^4-1$ by the minimal polynomial, and the remainder (which is lower degree) would be another polynomial relation that $A$ satisfied. – Aaron Dec 26 '14 at 13:37
@GeorgesElencwajg: yes sorry, the minimal polinomial of $A$ divides $x^4 -1 $, I've edited – WLOG Dec 26 '14 at 13:48
@WLOG but if I chose the matrix from above: $$A=\begin{pmatrix}0&-1\1&0\end{pmatrix}$$ I wouldn't get that $x^4-1$ is the minimal polynomial of $A$. In this case I would get that $x^2+1$ is the minimal polynomial of $A$. That's why I don't understand why you chose $x^4-1$ as the minimal polynomial of $A$. – SyndicatorBBB Dec 26 '14 at 14:01
@SyndicatorBBB: I don't choose $x^4 -1$ as minimal polynomial, I said that the minimal polynomial divides $x^4 -1$ – WLOG Dec 26 '14 at 14:02
Oh I'm sorry... Thank you. – SyndicatorBBB Dec 26 '14 at 14:05

Mher · Answer 3 · 2014-12-26T12:46:46.627

$\text{Hint (for the complex matrices):}$

actually your claim is true. If $p(A)=0$ where $p$ is a polynom with simple roots, then $A$ is diagonalizable. It follows from Jordan form of a matrix. Let $$A = Q^{-1}J_A Q=Q^{-1} \begin{pmatrix} J_A^1 & 0 &\dots& 0\\ 0& J_A^2&\dots& 0\\ &\dots&\dots\\ 0&0&\dots&J_A^n \end{pmatrix} Q$$

where $J_A^m, m=1,\dots,n$ are the Jordan cells of $J_A$: $$J_A^m= \begin{pmatrix} \lambda_m & 1 & 0 & \dots& 0\\ 0&\lambda_m & 1 & \dots& 0\\ &\dots& &\dots\\ 0 & 0 & 0 & \dots& \lambda_m\\ \end{pmatrix} $$

We claim that if $p(A)=0$ then all cells $J_A^m$ have the sizes $1\times 1$, otherwise $$p(A) = Q^{-1}p(J_A) Q=Q^{-1} \begin{pmatrix} p(J_A^1) & 0 &\dots& 0\\ 0& p(J_A^2)&\dots& 0\\ &\dots&\dots\\ 0&0&\dots&p(J_A^n) \end{pmatrix} Q$$

$$p(J_A^m)= \begin{pmatrix} p(\lambda_m) & \frac{p'(\lambda_m)}{1!} & \frac{p''(\lambda_m)}{2!} & \dots\\ 0&p(\lambda_m) & \frac{p'(\lambda_m)}{1!} & \dots\\ &\dots& &\dots\\ 0 & 0 & 0 & \dots& p(\lambda_m)\\ \end{pmatrix}\ne 0 $$ since $p$ doesn't have multiple roots: $p(\lambda)=p'(\lambda)=0$ has no solution.

In your case $p(x)=x^4-1$ has 4 distict roots.

I don't see why Jordan forms are needed here. Having an annihilator polynomial that splits into distinct monic factors of degree$~1$ suffices. — Marc van Leeuwen, Dec 29 '14 at 14:09

score 3 · Answer 4 · answered Dec 26 '14 at 12:31

3

The polynomial $P(x)=x^4-1$ annihilates $A$ so the minimal polynomial $\mu_A$ of $A$ divides $P$ so $\mu_A$ has simple roots so $A$ is diagonalizable.

answered Dec 26 '14 at 12:31

Why the minimal polynomial has only simple roots? – SyndicatorBBB Dec 26 '14 at 12:39
2

Because $P$ only has simple roots; any polynomial that divides $P$ must therefore only have roots of multiplicity at most 1. – Clement C. Dec 26 '14 at 12:44
Oh, He meant that $A$ is diagonalizable in $C$.... Thanks. – SyndicatorBBB Dec 26 '14 at 12:45
1

You should clarify over what underlying field this is diagonalizable. As other answers show, this is false for $\Bbb R$. – Pedro Dec 26 '14 at 13:50

score 1 · Answer 5 · answered Dec 29 '14 at 14:40

The term diagonalisable needs to be qualified by the field$~K$ over which one is working; for this reasons it is best to talk of diagonalisable linear operators (where $K$ is implicit in the vector space$~V$ that the operator acts upon) rather than matrices; the latter can be interpreted over various fields, and may be diagonalisable over some but not over others. The question stresses that that matrix has real entries, but that does not prevent it from being the matrix of a complex linear operator; I will however take this as an indication that being diagonalisable as a real linear operator is meant.

Given an annihilating polynomial for the linear operator, here $X^4-1$, one can always (in principle) factor it into pairwise relatively prime factors in $K[X]$, and obtain a direct sum decomposition of$~V$ into the kernels of those factors evaluated at the linear operator (or matrix). Here the finest such decomposition for $K=\Bbb R$ is $X^4-1=(X-1)(X+1)(X^2+1)$, so we get a decomposition $V=\ker(A-I)\oplus\ker(A+I)\oplus\ker(A^2+I)$. Each of these spaces might be $\{0\}$. When they are not, the first two spaces are the eigenspaces for $\lambda=1$ and $\lambda=-1$, respectively. However the third subspace does not contain any eigenvectors of$~A$, since the corresponding eigenvalue would have to satisfy $\lambda^2+1=0$ which is impossible for real values$~\lambda$. Having a nonzero dimensional $A$-stable subspace without eigenvectors ensures that $A$ is not diagonalisable (the restriction to a stable subspace of a diagonalisable operator is always diagonalisable), so one gets that $A$ satisfying $A^4=I$ is diagonalisable over$~\Bbb R$ if and only if $\ker(A^2+I)=\{0\}$. Equivalently, since this condition means that one has $V=\ker(A-I)\oplus\ker(A+I)$, it is diagonalisable if and only if $A^2=I$.

Just for completeness, such $A$ is always diagonalisable over $\Bbb C$, since the complex vector space $W$ acted upon decomposes $W=\ker(A-I)\oplus\ker(A+I)\oplus\ker(A-\mathbf iI)\oplus\ker(A+\mathbf iI)$, in which each of the summands is an eigenspace if it differs from$~\{0\}$.

If $A^4=I$ then $A$ must be diagonalizable?

5 Answers5