chapter 2 - Eigenvalues, Eigenvectors and Diagonalizability
Given a linear map can be represented by many different matrices, it's natural to ask if there a best choices of matrix representatives. Most square matrices can be diagonalized (as least working over the complex numbers) and the diagonal entries of such a matrix are called eigenvalues.
Chapter 3 - Spectral Theorem and Applications
Invertible changes of variable preserve algebraic properties but not geometric ones. An orthogonal change of variable preserves the inner product and so lengths, angles and areas. The spectral theorem shows that it is precisely the symmetric matrices which can be diagonalized via an orthogonal change of variable.
Given a linear map can be represented by many different matrices, it's natural to ask if there a best choices of matrix representatives. Most square matrices can be diagonalized (as least working over the complex numbers) and the diagonal entries of such a matrix are called eigenvalues.
Chapter 3 - Spectral Theorem and Applications
Invertible changes of variable preserve algebraic properties but not geometric ones. An orthogonal change of variable preserves the inner product and so lengths, angles and areas. The spectral theorem shows that it is precisely the symmetric matrices which can be diagonalized via an orthogonal change of variable.