Introduction

​​ Matrix

• Two dimensional array of numbers
• Denoted by bold capital letter
• A mathematical concept that is used to represent relationships between variables.

​​ Vectors

• A matrix with either one row or one column
• Denoted by bold lowercase letter

​​ Scalar

• A matrix with just one row and one column
• Denoted by any letter or symbol

​​ Square Matrix

• A matrix with the same number of rows as columns

​​ Diagonal Matrix

• A square matrix $a_{ij}$ where everything is zero except where $i=j$

​​ Identity Matrix

• A diagonal matrix $a_{ij}$ where all all $i = j$ is a 1

​​ J Matrix

• A matrix of any rows and column combination in which all elements equal 1

​​ Null Matrix

• A J matrix multiplied by 0

​​ Triangular Matrix

• Given a square matrix $a_{ij}$
  • A lower triangular matrix is when elements with $j>i$ are zero
  • An upper triangular matrix is when elements with $i>j$ are zero

​​ Tridiagonal Matrix

• A square matrix with all elements 0 except the diagonals and the elements to the left and right of said diagonals

Matrix Operations

​​ Transposition

​​ Diagonals

​​ Addition of Matrices

​​ Multiplication of Matrices

​​ Traces of Square Matrices

​​ Direct Sum of Matrices

​​ Kronecker Product

​​ Matrix Inversion

​​ Determinant of a Matrix

​​ Inverse of an Inverse