Matrices
1.1 Definitions
Definition of a matrix, element notation, size, transpose, and special matrix types: row, column, square, diagonal, identity, symmetric, anti-symmetric, triangular, null.
1.1 Definitions
Definition 1.1: An m × n matrix is a rectangular array of numbers arranged in m rows and n columns. Elements are denoted aij where i = row, j = column.
Example 1.1 — Various matrix sizes:
A (3×3):
| 1 | 2 | −5 |
| 0 | 3 | 2 |
| −1 | 5 | 0 |
B (4×1) column:
| 1 |
| 5 |
| −6 |
| 7 |
D (1×6) row:
| 0.9 | −.98 | 1.25 | 7.83 | 0 | 0.02 |
Key Definitions
Definition 1.2 — Row/Column matrix: A matrix with only one row is a row vector; with only one column is a column vector.
Definition 1.3 — Square matrix: m = n. The matrix A (3×3) above is square of order 3.
Definition 1.4 — Transpose AT: Rows and columns of A are interchanged. If C is 2×4, then CT is 4×2.
Example 1.2 — Transpose:
| 4.3 | −3.15 | 2.7 | 17.5 |
| 0 | 3 | −2.58 | −12.75 |
| 4.3 | 0 |
| −3.15 | 3 |
| 2.7 | −2.58 |
| 17.5 | −12.75 |
1.3 Special Square Matrices
Definition 1.10 — Principal diagonal: Elements aii from top-left to bottom-right.
Definition 1.11 — Diagonal matrix: dik = 0 for i ≠ k. Only main diagonal can be nonzero. Diagonal matrices commute: AB = BA.
Definition 1.12 — Identity matrix I: Diagonal matrix with all diagonal entries = 1. AI = IA = A for any compatible A.
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
Definition 1.13 — Symmetric: AT = A. Anti-symmetric: AT = −A (diagonal must be zero).
| 1 | 2 | −5 |
| 2 | 3 | 2 |
| −5 | 2 | 0 |
| 0 | 2 | −5 |
| −2 | 0 | 2 |
| 5 | −2 | 0 |
Definition 1.15 — Triangular: Upper-triangular: zeros below diagonal. Lower-triangular: zeros above diagonal.
| 2 | −7 | 8 |
| 0 | 5 | 15 |
| 0 | 0 | 5 |
| 5 | 0 |
| 4 | 5 |
Definition 1.16 — Null matrix: All elements are 0. Need not be square.