Finding the Inverse of a Diagonal Matrix in Terms of Eigenvalues

[P-Jay1]

Diagonalizing an N × N matrix H involves writing it as H = UDU† where D is a
diagonal matrix, with diagonal elements equal to the eigenvalues of the matrix H, and U
is a unitary matrix.
We may write:

D=
(λ1 0 0 ... 0)
(0 λ2 0 ... 0)
(0 0 λ3... 0)
(... ... ... ... λn)

Assuming all the eigenvalues are non-zero, how do I find an expression for the inverse matrix
D^−1 in terms of λi?
And how do I prove rove that H^−1 = UD^−1U†?


For the first question I'm assuming the inverse of D is just:

D=
(1/λ1 0 0 ... 0)
(0 1/λ2 0...0)
(0 0 1/λ3...0)
(... ... ... ...1/λn)

How do I find in terms of λi?

 

[HallsofIvy]

You don't need to "assume" the inverse of a diagonal matrix is the diagonal matrix having the reciprocal of each entry on the diagonal. Simply multiply the matrices and see what you get.

Of course, the inverse exists if and only if none of the diagonal entries is 0. And what you have already is precisely what is meant by "find an expression for the inverse matrix
$D^{−1}$ in terms of $λ_i$".

 

[P-Jay1]

Thanks Ivy. How do I go about doing the second question?