So, I understand how to orthogonally diagonalise a basic matrix with numbers in it. However, I have reached a question asking me to do so for a matrice involving only variables.
Orthogonally diagonalize $A = \pmatrix{a&b\\ b&a}$.
There is an answer at the back of the book, but I want to be able to understand the process.
My method is to first find the characteristic polynomial of $A$, which would involve factorising the determinant of $(A - \lambda I)$. But I can't factorise $\lambda^2 - 2a\lambda + (a^2 - b^2) = 0$.
Any ideas?
$\lambda^2 - 2a\lambda + (a^2 - b^2) = [\lambda-(a+b)][\lambda-(a-b)]$.