I'm having difficulty understand some questions. I will highlight the terms I do not understand.
Question 1:
Let $A =\begin{pmatrix} 1& -2 \\ 1& 3 \end{pmatrix}$
For the matrix $A$, decide whether there is an orthonormal basis $B$ for $\mathbb{C}^2$ such that the matrix of $A: \mathbb{C}^2 \rightarrow \mathbb{C}^2$ with respect to $B$ is diagonal. If so, find such a basis and write down a unitary matrix $P$ such that $\overline{P}^TAP$ is diagonal.
What I think we need to do: In class, we have learnt the following theorem -
"Spectral Theorem for normal operators"
Suppose that $V$ is an $n$-dimensional inner product space over $\mathbb{C}$ and that $T: V \rightarrow V$ is a linear map and is normal. Then there is an orthonormal basis for $V$ consisting of eigenvalues of $T$. In particular, if $A \in M_{n}(\mathbb{C})$ is normal then there is an unitary matrix $P \in M_{n}(\mathbb{C})$ such that $P^{*}AP$ is diagonal.
My guess is we need to apply this theorem in some way?
Now some terms I am having difficulty understanding is this sentence.
"Orthonormal basis $B$ for $\mathbb{C}^2$ such that the matrix of $A: \mathbb{C}^2 \rightarrow \mathbb{C}^2$ with respect to $B$". I understand what an orthonormal basis is but not sure about the rest of the sentence.
Question 2 (looks similar). This question is in an exam past test BUT not a continuation from the last question above.
Let $A =\begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix}$
For the matrix $A$, decide whether there is an orthonormal basis $B$ for $\mathbb{R}^2$ such that the matrix of $A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with respect to $B$ is diagonal. If so, find such a basis and write down an orthogonal matrix $P$ such that $P^TAP$ is diagonal.
So question two looks similar to 1) but we are asked for an orthogonal matrix instead of a unitary matrix. I understand what these two terms mean.
Other than that I have no clue on how to start these questions. If anyone can help me get started that would be great.
The matrix representation of A by using a basis which is not necessarily standard is $P^{-1}AP$, where P is some nonsingular matrix. Note that if there is some nonsingular P for which $P^{-1}AP=D$ where $D$ is a diagonal matrix, then $AP=PD$. Looking at each column, $Ap_i=\lambda_i p_i$, that is you get a basis for the space which consists of eigenvectors of $A$. Hence, what the questions are asking you to do is to exhibit a special basis consisting of eigenvectors of $A$ (special because you have to realize if $P^{*}AP=D$ is possible in the first place. Spectral Theorem answers this via the normality of $A$) that will diagonalize $A$. If you have methods in your class doing this, then you should be able to apply them here.