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April 23rd, 2018, 02:08 PM   #1
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Matrices which satisfy the equation

Show that there are no real 3x3 matrices which satisfy the equation (picture below), but there are complex 3x3 matrices and real 2x2 matrices which satisfy that.
I know that this equation has no real roots, but I don't know how to apply that to matrices.
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April 23rd, 2018, 02:12 PM   #2
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do you mean that $\lambda^2 + 1$ is the characteristic equation of the matrix?
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April 23rd, 2018, 02:13 PM   #3
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Quote:
Originally Posted by romsek View Post
do you mean that $\lambda^2 + 1$ is the characteristic equation of the matrix?
Yes
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April 23rd, 2018, 03:51 PM   #4
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Quote:
Originally Posted by Birgitta View Post
Yes
sorry I meant characteristic polynomial
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April 23rd, 2018, 03:53 PM   #5
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sorry I meant characteristic polynomial
I assumed that. However, I've already solved the problem. Thanks
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April 23rd, 2018, 05:03 PM   #6
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In a nutshell, the main idea is to prove the fact that if $M$ is a matrix defined over a field, $F$, and it satisfies $\lambda^2 + 1 = 0$, then every vector in $F^3$ is a (generalized) eigenvector for either $i$ or $-i$. Note this is NOT saying that $\lambda^2 + 1$ is the characteristic polynomial for $M$ (though it is true that it must be a multiple of $\lambda^2 + 1$.
EDIT: After thinking about it, I realized even this claim is not true. $\lambda^2 + 1$ need only have one linear factor in common with the characteristic polynomial. It's best to avoid thinking about the characteristic polynomial at all since it has nothing to do with the question.

With this fact in hand, it should be trivial to find examples for $\mathbb{R}^2$ which work as well as examples in $\mathbb{C}^3$. I'll outline the proof below which should give you a hint.

1. Suppose $M$ is a 3x3 matrix and $M^2 + I = 0$. Then if $F = \mathbb{R}$, then $i,-i$ are eigenvalues for $M$ and moreover, they correspond to a pair of linearly independent eigenvectors. Denote these (unit vectors) by $\xi_1,\xi_2$ and note that they are conjugates (prove this and note that already this may fail if $F = \mathbb{C}$).

2. Prove that $\xi_1,\xi_2$ span a real invariant subspace with respect to $M$. Hint: prove that $\xi_1 + \xi_2$ and $\xi_1 - \xi_2$ span a 2-dimensional subspace of $\mathbb{R}^3$. Let $V = \text{span} \{\xi_1,\xi_2 \}$ denote this subspace.

3. Choose any vector $u \in \mathbb{R}^3 \setminus V$ and consider the action of $M$ on $u$ expanded in the basis $\{ \xi_1,\xi_2, u\}$ as
\[ Mu = a_1 \xi_1 + a_2 \xi_2 + a_3 \xi_3. \]
Now, $a_3 \in F$ and $a_3 \neq 0$ (why not?). Note that $a_1,a_2$ may be in $\mathbb{C}$ since I am using $\xi_1,\xi_2$ in my basis. However, this isn't a problem if you have proved (2).

4. Note that $M^2u = -u$ by assumption but also from (3),
\[M^2u = a_1 i \xi_1 - a_2 i \xi_2 + a_3 Mu.\]
From this you can argue three things:
(i) $a_1 = a_2 = 0$.
(ii) $u$ is a nonzero eigenvector for $M$ with eigenvalue $a_3$.
(iii) $a_3$ satisfies $a_3^2 = -1$

5. From (4) you can conclude that such an $M$ exists only if $a_3 = \pm i$ but that requires $F = \mathbb{C}$ since you proved in (3) that $a_3 \in F$.
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Last edited by skipjack; May 2nd, 2018 at 03:07 PM.
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May 2nd, 2018, 02:14 PM   #7
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A 3x3 matrix has to have a $\displaystyle \lambda^{3}$ term.

For a 2x2 matrix, the characteristic equation is
$\displaystyle \lambda^{2} - \lambda (a_{11}+a_{22})+D = 0$
and you can easily pick real components to satisfy OP.
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May 2nd, 2018, 07:23 PM   #8
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Quote:
Originally Posted by zylo View Post
A 3x3 matrix has to have a $\displaystyle \lambda^{3}$ term.

For a 2x2 matrix, the characteristic equation is
$\displaystyle \lambda^{2} - \lambda (a_{11}+a_{22})+D = 0$
and you can easily pick real components to satisfy OP.
1. This has nothing to do with the characteristic equation.

2. If what you are saying was true, the very thing the OP is supposed to prove would be false.
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May 3rd, 2018, 06:33 AM   #9
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Quote:
Originally Posted by zylo View Post
A 3x3 matrix has to have a $\displaystyle \lambda^{3}$ term.

For a 2x2 matrix, the characteristic equation is
$\displaystyle \lambda^{2} - \lambda (a_{11}+a_{22})+D = 0$
and you can easily pick real components to satisfy OP.
Thought it was clear: The characteristic polynomial of a 3x3 matrix has to be of degree 3. OP is of degree 2.
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May 3rd, 2018, 07:04 AM   #10
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Quote:
Originally Posted by zylo View Post
Thought it was clear: The characteristic polynomial of a 3x3 matrix has to be of degree 3. OP is of degree 2.
There is no reason why the $\lambda^2+1$ in the OP is the characteristic polynomial. In fact, it isn't for a $3\times 3$-matrix.
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