Lecture 7: Introduction to Matrices

Introduction to Matrices

Context: In previous lectures, systems of linear equations and elimination methods were studied.
Purpose of Matrices: To work with these systems more efficiently and compactly.
Functionality: A matrix provides a structured way to organize numbers, specifically coefficients and constants, into rows and columns.
Foundation: Understanding the structure of matrices is fundamental to all of linear algebra.

Definition: A matrix is formally defined as a rectangular array of numbers.
Components:
- Rows: Horizontal arrangements of numbers. Example: [\begin{matrix} 1 & 2 & 3 & 4 \end{matrix}]
- Columns: Vertical arrangements of numbers. Example: [\begin{matrix} 1 \ 5 \ 9 \end{matrix}]

Definition: The size of a matrix is determined by the number of its rows and columns.
- If a matrix has m rows and n columns, its size is denoted as m \times n (read as "m by n").
Example: A 3 \times 4 matrix has 3 rows and 4 columns.
Notation: A matrix is often denoted as A = [a{ij}], where a{ij} represents the entry located in the i-th row and the j-th column.

Definition: An m \times n matrix is considered square if the number of rows equals the number of columns, i.e., m = n.
General Form: A square matrix has a symmetric number of rows and columns.
Example (A 3 \times 3 square matrix):
\begin{pmatrix} a{11} & a{12} & a{13} \ a{21} & a{22} & a{23} \ a{31} & a{32} & a_{33} \end{pmatrix}

Definition: Two matrices, A and B, are defined as equal if and only if two conditions are met:
1. Same Size: They must have identical dimensions (same number of rows and columns, m \times n).
2. Matching Entries: Each corresponding entry must be equal, meaning a{ij} = b{ij} for all possible values of i and j.
Example 1 (Equal Matrices):
- \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} and \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} are equal because they have the same size and every corresponding entry matches.
Example 2 (Illustrating non-equality or solving for equality):
- If \begin{pmatrix} x & 2 \ 3 & y \end{pmatrix} = \begin{pmatrix} 5 & 2 \ 3 & 7 \end{pmatrix}, then by the definition of equality, this implies the equations x = 5 and y = 7. Each entry must match its counterpart for the matrices to be equal.

1. Zero Matrix (denoted by O):
- Definition: A matrix in which all entries are zero (0).
- Entry Notation: a_{ij} = 0 for all i, j.
- Example:
  \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}
2. Identity Matrix (denoted by I):
- Definition: A square matrix characterized by ones on its main diagonal (from top-left to bottom-right) and zeros everywhere else.
- Entry Notation: In entry notation, I = [a{ij}] where a{ij} = 1 if i = j (on the diagonal) and a{ij} = 0 if i \neq j (off the diagonal). This can be compactly written using the Kronecker delta notation as a{ij} = \delta_{ij}.
- Example (A 3 \times 3 identity matrix):
  \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}

Definition of Matrix: A rectangular array of numbers.
Orientation: Rows are horizontal; columns are vertical.
Size (Shape) of a Matrix: Given by m \times n, where m is the number of rows and n is the number of columns.
Square Matrix: A matrix where the number of rows equals the number of columns (m=n).
Equality of Matrices: Two matrices are equal if they possess the same size and all their corresponding entries are identical.
Special Matrices:
- Zero Matrix (O): All entries are 0.
- Identity Matrix (I): A square matrix with 1's on the main diagonal and 0's elsewhere.