MATH 462, Review for Midterm Test 2

Test topics

1. Determinants
   1. The area function and determinant (4.1)
   2. Definition of the determinant and the basic properties (4.2)
   3. Characterization of the determinant (4.5: Theorem 4.12)
   4. Determinants of matrix products and transposes (4.3)
2. Diagonalization
   1. Eigenvalues and eigenvectors (5.1)
   2. Diagonalization (5.2)
   3. Invariant subspaces and the Caley-Hamilton theorem (5.4)
3. Canonical forms
   1. The Jordan canonical form (7.1)

Important definitions

• Determinant of a matrix (cofactor expansion)
• Multilinear (n-linear), alternating functions
• Eigenvectors, eigenvalues, eigenspace
• Diagonalizable operator
• Characteristic polynomial, characteristic equation
• Invariant subspace, cyclic subspace
• Generalized eigenvectors, eigenspaces
• Cycle of generalized eigenvectors
• The Jordan canonical form, the Jordan basis

Important theorems

• Characterization of determinant (Theorem 4.12 and related facts)
• Determinants of matrix products and transposes (Theorems 4.7, 4.8)
• Criterion for diagonalizability (Theorem 5.1)
• Linear independence of eigenvectors (Theorem 5.5)
• The algebraic and geometric multiplicity (Theorem 5.7)
• Invariant subspaces and characteristic polynomials (Theorem 5.21)
• Basis of a cyclic subspace (Theorem 5.22)
• The Caley-Hamilton theorem (Theorem 5.23)
• The vector space V is a direct sum of generalized eigenspaces (Theorem 7.3)
• A generalized eigenspace has a basis which is a union of cycles (Theorem 7.7)

A number of questions on the test will be simple yes/no type questions, which may be taken from Problems 1 in each section.