Course Description:
This course covers eigenvalues and eigenvectors, system of linear equations, orthogonality, direct sum and decomposition of vector spaces and canonical forms
Course objectives
On the completion of the course, successful students will be able to:
|
Find the eigenvalues and eigenvectors of a square matrix, Identify similar matrices Diagonalize matrix when it is possible Define inner product space Find and apply the LU factorization of a matrix |
Understand the Gram-Schmidt process Find an orthogonal basis for a subspace Find an orthogonal complement of a subspace Recognize and invert orthogonal matrices Comprehend the three canonical forms of matrices |
Chapter one: the Characteristic equation of a matrix
1.1. Eigenvalues and eigenvectors
1.2. The characterstic polynomial
1.3. Similarity of matrices and characterstic polynomial
1.4. The spectral radius of a matrix
1.5. Diagonilization
1.6. Decomposable and Cayley-Hamilton theorem
Chapter Two: Orthogonality
2.1. The inner product
2.2. Inner product spaces
2.3. Orthonormal sets
2.4. The gram Schmidt orthogonilization process
2.5. Cauchy-Schwartz and triangular inequalities
2.6. The dual space
2.7. Adjoint of linear operator
2.8. Self-adjoint linear operators
2.9. Isometry
2.10. Normal operators and the spectral theorem
2.11. Factorization of a matrix (LU, Cholesky, QR)
2.12. Singular value decomposition
Chapter three: Canonical forms
3.1. Elementary row and column operations on matrices
3.2. Equivalence of matrices of polynomial
3.3. Smith canonical forms and invariant factors
3.4. Similarity of matrices and invariant factors
3.5. The rational canonical forms
3.6. Elementary divisors
3.7. The normal and Jordan canonical forms
Chapter four: Bilinear and quadratic forms
4.1. Bilinear forms and matrices
4.2. Symmetric bilinear forms and quadratic forms
4.3. Real symmetric bilinear forms, positive definite forms
Chapter five: Direct sum decomposition of vector spaces
5.1. Definition o direct sum of vector spaces
5.2. Projection and invariant subspaces of a linear operator
5.3. Primary decomposition theorem
- Teacher: ZELALEM MESERET