# Thread: Find the values of "z" for which this Matrix has no real eigenvalues

1. ## Find the values of "z" for which this Matrix has no real eigenvalues

I got this problem:

Find the values of "z" for which this Matrix has no real eigenvalues.
The Matrix is the following:
[ 1 -1]
[ 1 z]

I tried to solve this problem as follows:
1) I calculated the eigenvalues for the Matrix keeping "z" as a constant
2) I got:
-1/z1/2 and 1/z1/2
So if I substitute z with >0 I have real eigenvalues.
For z <0 I don't have real eigenvalues.

But what happens with the 0? If I substitute 0 for z the result is not defined, but if I substitute 0 for z in the matrix I get two real eigenvalues (0,0).

Therefore I say that the Matrix has no real eigenvalues for z<0

Is this true?

2. ## Re: Find the values of "z" for which this Matrix has no real eigenvalues

Some thing is going wrong!

The eigenvalues are the sol of the following equation
$(1-\lambda)(z-\lambda)+1=0$
that is
$\lambda^2-\lambda(1+z)+(1+z)=0$
so this equation has a real valued solutions if and only if
$(1+z)^2-4(1+z)\geq0$
which equivalently means
$-1\leq z\leq3.$

3. ## Re: Find the values of "z" for which this Matrix has no real eigenvalues

But if I set for example z=5, than I get real eigenvalues: (1,2679 4,7321) - therefore it has real eigenvalues also for z=5 or not?

4. ## Re: Find the values of "z" for which this Matrix has no real eigenvalues

Sorry! the eigenvalues are real when z is out of the interval $-1.
So, we have real eigenvalues if and only if $z\geq3$ or $z\leq-1$

5. ## Re: Find the values of "z" for which this Matrix has no real eigenvalues

Kmath corrected while I was typing this!