# Matrix Analysis Resources

## LECTURE NOTES

CAUTION: Except for the syllabus above, all the material provided on this web page is course material of one professor (Zweck). The exam however will be written by a committee of professors!

Here are lecture notes from the last time I taught Math 430. Apologies for my poor handwriting, lines cut off at ends of pages by scanner, and any mistakes.

These notes are mostly based on the book of Meyer. They are provided to give students more of an idea of the sort of material that will be covered on the qualifying exam.

Note that two topics in the qualifying exam syllabus are not covered in these notes: Generalized Inverses (i.e., Moore-Penrose pseudoinverse) and the Jordan Canonical Form. Also, material from Lecture 14 below is not included in the qualifying exam syllabus.

Lecture 1: Overview
Lecture 2: Matrix Algebra
Lecture 3: Vector Spaces and Subspaces; Range and Nullspace
Lecture 4: Linear Independence, basis, dimension; Rank and Nullity Theorem
Lecture 5: Least Squares
Lecture 6: Linear Transformations; Change of basis, similarity
Lecture 7: Orthogonality, Orthonormal basis, Gram-Schmidt
Lecture 8: Orthogonal and Unitary Matrices
Lecture 9: Complementary Subspaces; Orthogonal decomposition
Lecture 10A: Block Gauss-Jordan Elimination, Schur Complements
Lecture 10B: Determinants
Lecture 11: Eigensystems and Differential Equations
Lecture 12: Diagonalization
Lecture 13: Spectral Theorems for Normal and for Symmetric Matricies
Lecture 14: Functions of Diagonalizable Matrices; Applications to Systems of ODEs
Lecture 15: Quadratic Forms and Positive Definite Matrices
Lecture 16: Nilpotent Matrices and Jordan Blocks
Lecture 17: Range-Nullspace Decomposition
Lecture 18: Jordan Canonical Form

## PAST EXAMS

Here are exams (some with solutions) I have given in Math 430. Note that the material covered in these exams does not exactly overlap with that to be covered on the Matrix Analysis Qualifying exam. Moreover, the format, degree of difficulty, and choice of topics of the Qualifying Exam will be somewhat different to that exhibited in these past Math 430 exams.

## Exams from Matrix Analysis in Fall 2005

Exam One [pdf]
Exam One Solutions [pdf]
Exam Two [pdf]
Exam Two Solutions [pdf]
Final [pdf]

## Exams from Matrix Analysis in Fall 2006

Exam 1 [pdf]
Exam 1 Solutions [pdf]
Exam 2 [pdf]
Exam 2 Solutions [pdf]
Final Exam [pdf]

## Exams from Matrix Analysis in Fall 2008

Exam One [pdf]
Solutions to Exam One [pdf]
Solutions to Exam Two [pdf]
Final Exam [pdf]
Solutions to Final Exam [tif]