Mathematics for AI 1

(1) Purpose and content of the course

In this course, participants will learn the fundamentals of matrices, which are an indispensable element in learning machine learning. This course is intended for beginners who have no experience in learning linear algebra, including matrices, in classes, etc. Therefore, the course is designed for those who have no experience in learning matrices or linear algebra. Therefore, the course starts with an introduction to the definitions of “matrix” and “vector,” and then explains how to calculate matrices and how to use them with concrete examples. The ultimate goal of this course is to enable students to understand the diagonalization of matrices and to understand linear algebra necessary for learning machine learning.

(2) Contents of each session

Session 1: Introduction, Basics of Matrices (1)

 The course aims to explain the definitions and calculation methods of vectors and matrices, and to enable students to perform basic matrix calculations (sums and products).

Session 2: Fundamentals of Matrices (2)

Definition of unitary matrix and inverse matrix

Relationship between simultaneous equations and inverse matrices

How to find matrices by the sweep method

The simultaneous equations learned in junior high school can be written as matrix calculations. Finding solutions to simultaneous equations using matrices is one of the most basic uses of matrices, and is explained in this section. The students will learn to solve simultaneous equations systematically by using a method called the sweep method.

Session 3 Eigenvalues 

The definition of eigenvalues and how to obtain them will be explained. Eigenvalue is one of the most fundamental features of matrices and is an important concept that appears not only in machine learning but in all fields. Students will understand the meaning of eigenvalues and how to obtain them.

Session 4: Diagonalization

This lecture explains the diagonalization of matrices, a very important matrix transformation method, as we proceed with the theory of eigenvalues. Some matrices can be diagonalized and some cannot, and the difference between the two will be explained. As an application of diagonalization, a method to obtain powers of a matrix will also be explained.

Session 5: Jordan Standard System

For non-diagonalizable matrices, there is the Jordan standard system, which is similar to the diagonal matrix. This lecture is somewhat advanced, but it is important to learn it because it appears not only in machine learning but also in many other situations.