Le that $(a_i)^{n}_{i=1}$ be an orthonormal basis and $C(x)$ be a transformation defined as follows:
$$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k \rangle\\ \vdots \\ \langle x, a_n \rangle\\ \end{array} \right)$$
is there a way to express that (linear) function with matrix multiplication?
Note that $a_i$ makes an orthonormal basis and each $a_i \in \mathbb{C}^d$ and $\langle a, b \rangle = a^*b = a^H b$.
i.e. what I am looking for looks something like this:
$$C(x) = Cx = \tilde{x}$$
where my intuition says C should be a unitary matrix in terms of the basis $a_i$.
This is what I have tried:
It seems to me that the only way to do this is to define C as follows. Let the columns of C be $a_i$. Then take the matrix product of x and the hermitian C i.e. let C(x) = :
$$C^*x = \tilde{x}$$
which doesn't seem entirely correct to me.
Let me tell you why I think its not right. First in this context, $\langle a, b \rangle = a^Hb = a^*b$ and hence $\langle a, b \rangle \neq \langle b, a \rangle $ in general.
$$C^*x = = \left( \begin{array}{c} a_1\\ \vdots \\ a_k\\ \vdots \\ a_n\\ \end{array} \right) x = \left( \begin{array}{c} \langle a_1, x \rangle\\ \vdots \\ \langle a_k, x \rangle\\ \vdots \\ \langle a_n, x \rangle\\ \end{array} \right) $$
and as I pointed out $ \langle a_k, x \rangle \neq \langle x , a_k \rangle $. So my answer is wrong ... but close to what I wanted, it just has the inner products the wrong way round.
However, it seems that in the real case $C^* = C$, so that would be right answer...no? Since in the real case the inner product $\langle a , b \rangle$ is a symmetric function.. is that right in this case?
Thought it doesn't answer my original question though.
I was expecting C(x) to have the following properties:
- C is a linear function
- it can be identified with d by d matrix with columns given by the orthonormal basis
- it is a unitary transformation ( $C^*C = I$)
1 seems obvious to me because we need $C(x + y) = C(x) + C(y)$ which seems clear it obeys it because of the properties of inner products.
2 and 3 are the ones I am having a hard time obtaining.
As an example, I was told the Fourier transform works with the following basis:
$a_k = \frac{1}{\sqrt{d} }(1, e^{-2 \pi i j \frac{1}{d}} , \dots, e^{-2 \pi i j \frac{g}{d}} , \dots, e^{-2 \pi i j \frac{d-1}{d}} )$
with $a_k$ as the columns, but I am skeptical because I can't make it work for a general orthonormal basis.
Any help is appreciated.
As $\langle x, a \rangle$ is conjugate-linear in $x$, rather than linear, there's no way you can get that by a matrix multiplication, which is a linear operator.