Weeks 13 and 14 -- Sections 8.1-8.2

Short-term objectives

1. Find the eigenvalues and eigenvectors of a matrix. (Sec. 8.1)
2. Diagonalize a matrix: Find the related diagonal matrix and the change-of-basis matrix (similarity transformation) relating them. (Sec. 8.1)
3. Recognize when an orthonormal eigenbasis exists, and when no eigenbasis exists at all. (Sec. 8.1, 8.2)
4. Calculate functions of an operator or matrix, and use the exponential function to solve systems of differential equations. (Sec. 8.1)
5. 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

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