How can this be equal to the unit matrix?

[Reid]

Homework Statement

At the lecture yesterday the teacher just ended up with a result I could not arrive at. So, how
can the below stated expression be verified?

$$\left(C^{1/2}\right)^{T}C^{-1}C^{1/2}=I$$
Here C is a nonsingular covariance matrix, obviously, and I is the unit matrix.

I will not make an attempt of a solution because then it feels like I would solve it but not understand. I hope that is ok. What I seek here is not an rigorous proof. I just want to understand.

Hope someone can help me!

 

[D H]

Suppose some invertible matrix n×n C is decomposed into the product of two n×n matrices A and B:

$$C = AB$$

Then A and B must themselves be invertible and the inverse of C is given by

$$C^{-1} = B^{-1}A^{-1}$$

The definition of the matrix square root of some matrix C is that

$$C=\left(C^{1/2}\right)^T\,C^{1/2}$$

Combine the above two and the result in the original post falls right out.

 

[Reid]

If I understand you correctly, then it is allowed to change the order in the matrix multiplication?
$$C^{-1}C^{1/2}=C^{1/2}C^{-1}$$

 

[Reid]

There was no need to change the order. Now I see. :) Thanks!