Eigenvalues - real and imaginary

[jpildave]

Am I understanding this right?

Let's say I have a 15x15 matrix called Z. Then the matrix of eigenvalues calculated from Z, called D, can have two forms - either diagonal or block diagonal.

If the matrix D comes out with values only on the diagonal, then there are only real values. But, if the matrix D comes out block diagonal, then there are real and imaginary values.

The only reason the matrix D comes out block diagonal is if it is not symmetric, not only in terms of dimensions, but in terms of the actual values in the original matrix Z.

Correct?

 

[LogicalTime]

I believe diagonal is a subset of Jordan block diagonal, since you can count one number as a Jordan block.


Is each entry in Z a real number or complex?


$\left[ \begin{array}{rr} 0&-1\\ 1&0\\ \end{array} \right]$
has eigenvalues = i, -i