For a Linear Transformation from $\mathbb{R^4} \to \mathbb{R^4}$ which option is correct?

Question

Consider the Linear Transformation on $\mathbb{R^4} \to \mathbb{R^4}$ given by the matrix

$$ A = \begin{pmatrix} 1 & 2 & 0 & 0\\ 0 &2 & 0 &0\\ 0 &0 &0 &1\\ 0 &0 &-1 &0 \end{pmatrix}$$ Then the number of Linearly Independent Vectors whose direction is invariant under this transformation is :

(a) 0

(b) 1

(c) 2

(d) 4

Now, this question is simply asking for the number of L.I. Eigen Vector corresponding to the given Linear Transformation.

Now, the eigen values are given by : $1,2,i, -i$

Now, I know this theorem that eigen vectors corresponding to distinct eigen values are Linearly Independent.

So, $4$ should be the correct answer .

But, if all $4$ eigen vectors are L.I. then the matrix is diagonalisable.

Although for matrix to be diagonalisable the eigen values must belong to the field in which we are working.

(This is the precise reason why the matrix $\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$ is not diagonalisable over $\mathbb{R}$ although its eigen values are distinct.)

So, My question is :

What exactly is the correct answer to this question is it option (c) because only $2$ eigen values belong to the field $\mathbb{R}$ or is it option (d)

Can someone help me here ?

Thank you.

Answer 1

The answer is $2$. In this context, the only eigenvalues that matter are the real ones, and there are $2$ real eigenvalues. You can take, say, the vectors $(1,0,0,0)$ and $(2,1,0,0)$.