Course

MATH-115(a): Advanced linear algebra II - diagonalization

Lectures in this course (24)

Covers symmetric matrices, eigenvalues, and diagonalization process for spectral theorem applications.

Covers positive definite matrices and their properties, focusing on strict positivity of principal minors.

Explores the Spectral Theorem, emphasizing the Min-Max Criterion for symmetric matrices and the properties of positive definite matrices.

Covers Singular Value Decomposition and Pseudoinverse, explaining their applications in data compression and linear systems.

Explores finding the minimal solution of a problem using orthogonal bases and factorization, emphasizing uniqueness and practical examples.

Explores low rank approximations, spectral theorems, and orthogonality in matrices.

Explores linear algebra over integers, emphasizing solutions to A.x=b equations and Hermite normal form.

Explores the exponential of a matrix, properties, Frobenius norm, nilpotent matrices, commutativity, and system solutions.

Explores polynomial factorization over a field, emphasizing eigenvalues and irreducible components.

Covers the factorization of polynomials over a field, including division with remainder and common divisors.

Explores finitely generated abelian groups, focusing on their structure and isomorphism theorems.

Covers the Hermite Normal Form, a method to transform a matrix into a specific form.

Covers the Jordan Normal Form and its application in linear algebra.

Explores the conditions and implications of diagonalizability of linear transformations, including eigenvectors, eigenvalues, and distinct matrices.

Covers the diagonalization of matrices with distinct eigenvalues and the importance of this process.

Covers Cramer's Rule for solving linear equations using determinants.

Covers bilinear forms in vector spaces, their properties, orthogonality, and equivalence conditions.

Covers the basics of polynomials, focusing on rings, operations, and properties.

Covers polynomial roots, factorization, and unique representation through examples of polynomial division with remainders.

Covers the matrix exponential, its convergence properties, and the Jordan normal form.