Principal axes of a matrix

Question

I try to find the definition of the main axes of a matrix. I saw this phrase in some exercise:

Let $A$ be a positive matrix, $f:G\longrightarrow \mathbb{R}$ a smooth function, $G$ an open set in $\mathbb{R}^n$. I need to find the orthogonal coordinate transformation $y=Px$ such that the main axis on $y$'s coordinates will be the principle axis of $A$.

The book says to diagonalize $A$: $PAP^t=D$ and to choose $P$ to be the transformation.

What is the definition of principal axes of matrix?

Answer 1

Often, principal axes of a matrix refer to its eigenvectors. With this diagonalization, $P$ is the matrix of eigenvectors.