A is similar to B if there exists a matrix P such that P−1AP=B
A is diagonalizable if it is similar to a diagonal matrix D.
Suppose we want P to be an orthogonal matrix, not just any invertible matrix this give rise to the following definition
Definition
We say that two nxn matrices A and B are orthogonally similar if there exists an orthogonal matrix P such that PTAP=B
we say that A is orthogonally diagonalizable if A is similar to diagonal matrix D. that is there exists an orthogonal matrix P and diagonal matrix D such that:
PTAP=D
When is a matrix orthogonally diagonalizable?
Let A be matrix ok! to know if a matirx is orthoganlly diagonalizable, we must find if the dot products of the eigenspace is 0
How to orthogonally diagonalize
Its less about how to do it, but what the relation ship looks like to the vector
You might be asked to find PTAP=D or in the terms of A=PDPT