Write $A$ as a matrix in the $|a_i\rangle$ basis

Question

Suppose $A$ is a Hermitian operator with eigenvalues $a_1$, $a_2$, and $a_3$, and eigenvectors $|a_1\rangle$, $|a_2\rangle$, and $|a_3\rangle$, where eigenvector $|a_i\rangle$ corresponds to eigenvalue $a_i$.

How does one write $A$ as a matrix in the $|a_i\rangle$ basis? I know you can express $A$ as $$\sum_i a_i \langle a_i |a_i\rangle,$$

But this doesn't help me express it as a matrix. Anyone know what I can do?

Answer 1

The representation in this basis is

$$ A_{ij} = \langle a_i| A|a_j\rangle = a_j\langle a_i|a _j\rangle $$

if the basis is orthonormal then

$$ A_{ij} = a_j\langle a_i|a _j\rangle = a_j\delta_{ij} $$

Answer 2

Hint: Writing an operator w.r.t. some basis is asking the question: "by which Matrix is the linear transformation of coordinate vectors of that basis described?". Consider $v=\lambda_1 |a_1\rangle+ \lambda_2 |a_2\rangle + \lambda_3 |a_3\rangle$ and find out what the coordinates of $Av$ are w.r.t. your basis.

Note that this approach (writing a linear operator as a matrix) is valid for operators in finite-dimensional vector spaces.