I have been thinking this exercise for a while and I am really clueless about how to construct the basis of the eigenspaces . ( I point out as always, that english is not my language, though I manage to translate the exercise from spanish and understand the answers you give)
Exercise: Considering the canonical inner product, let T be an linear operator, $T:{R}^{4} \mapsto {R}^{4}$ and A and B non trivial subspaces such as:
$T_{ | A}=Id $ , $T_{ | B}= -Id , $ , $ N\left ( T-2Id \right ) = \left ( A+B \right )^{\perp }$ and $rank\left ( T-2Id \right )=2$
- Find a basis for ${R}^{4}$ consisting of eigenvectors of T
- Prove T is invertible
- Let $S:{R}^{4}\mapsto {R}^{4}$ such as $S=T^{3}+4T^{-1}+2Id$ . Prove S is diagonalizable and find the eigenvalues.
- If $\left \langle x,z \right \rangle=0, \forall x \in X, \forall x \in Z$, could it be possible to find an orthonormal basis of $T:{R}^{4}$ consisting of eigenvectors of T?
Here my thoughts:
- A has at least dimension 1, and is associated to T eigenvalue 1
- B has at least dimension 1, and is associated to T eigenvalue -1
- T also has eigenvalue 2 with geometric multiplicity 2
- Based on the assumptions above I say that T is associated to this diagonal matrix D= $\begin{bmatrix}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}$
- As the matrix above is non singular, T is invertible.
- It is easy to find S and it s diagonal matrix, with eigenvalues 12, 12, 7 and -3.
I still dont have the slightless idea of how to complete items 1 and 4 or how to relate this to any inner product.
Could somebody be so kind to give me a clue ? I am studying on my own and sitting next saturday for a test