Matrix diagonalisation with complex eigenvalues

[misterau]

Homework Statement

Is there a basis of R4 consisting of eigenvectors for A matrix?
If so, is the matrix A diagonalisable? Diagonalise A, if this is possible. If A is not diagonalisable because some eigenvalues are complex, then find a 'block' diagonalisation
of A, involving a 2 × 2 block corresponding to a pair of complex-conjugate eigenvalues.

A=
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0

Homework Equations

The Attempt at a Solution

I worked out the eigenvalues 1,-1,i,-i. I also worked out real eigenvectors: (-1,1,-1,1)^T and (1,1,1,1)^T. Whenever I try to work out complex eigenvectors I get no free variables..? How come A is "not diagonalisable because some eigenvalues are complex?" What does it mean by find block disgonalisation?

 

[VKint]

A is diagonalizable, just not over \mathbb{R}. As for the block-diagonalization, what they mean is that given a real 2 \times 2 matrix B with eigenvalues \alpha \pm i \beta, there exists a real, nonsingular matrix S such that
<br /> S^{-1} B S = \begin{pmatrix}<br /> \alpha &amp; \beta\\<br /> -\beta &amp; \alpha<br /> \end{pmatrix} \textrm{.}<br />
This is sometimes called the real canonical form of B. The way you find the matrix S is by first finding some complex eigenvector of B corresponding to the eigenvalue \alpha + i \beta, say \mathbf{w} = \mathbf{u} + i\mathbf{v}, where \mathbf{u} and \mathbf{v} are real vectors. Then S is the matrix with columns \mathbf{u} and \mathbf{v}.

In your case, the real canonical form of the matrix will turn out to be
<br /> \begin{pmatrix}<br /> 1 &amp;0&amp;0&amp;0\\<br /> 0&amp; -1 &amp;0&amp;0\\<br /> 0&amp;0&amp;0&amp; 1\\<br /> 0&amp;0&amp; -1 &amp;0\\<br /> \end{pmatrix} \textrm{.}<br />
This is what is meant by block-diagonalization. A basis that puts your original matrix into this form will be \{ \mathbf{w}_{+1}, \mathbf{w}_{-1}, \mathbf{u}, \mathbf{v} \}, where \mathbf{w}_{\lambda} indicates an eigenvector corresponding to the eigenvalue \lambda and \mathbf{u} + i \mathbf{v} is an eigenvector with eigenvalue + i.