### Weeks 13 and 14 -- Sections 8.1-8.2

#### Short-term objectives

- Find the eigenvalues and eigenvectors of a matrix. (Sec. 8.1)
- Diagonalize a matrix: Find the related diagonal matrix and the
change-of-basis matrix (similarity transformation) relating them.
(Sec. 8.1)
- Recognize when an orthonormal eigenbasis exists, and when no eigenbasis
exists at all. (Sec. 8.1, 8.2)
- Calculate functions of an operator or matrix, and use the exponential
function to solve systems of differential equations. (Sec. 8.1)
- Use the signs of the eigenvalues of a quadratic form to test a function
of several variables for extrema. (Sec. 8.2)

#### Long-term objectives

- Appreciate the role of the eigenvector concept in solving ordinary
and partial differential equations. (Sec. 8.1)
- Understand the various definitions of "orthogonal matrix" and why they
are equivalent. (Sec. 8.2)
- Understand the special properties of symmetric real matrices, and follow
the simpler proofs. (Sec. 8.2)
- Recognize finite-dimensional instances of the "Fredholm alternative".
(Sec. 8.2)
- Understand the relationship among conic sections, quadratic forms, and
spectral invariants (especially trace and determinant). (Sec. 8.2)
- Know the zeroth- through second-order terms in the Taylor expansion
of a function of several variables. (Sec. 8.2)