Matrices and Operations

Last updated: December 2024

Disclaimer: These are my personal notes compiled for my own reference and learning. They may contain errors, incomplete information, or personal interpretations. While I strive for accuracy, these notes are not peer-reviewed and should not be considered authoritative sources. Please consult official textbooks, research papers, or other reliable sources for academic or professional purposes.

1. Matrix Definition

A matrix $A$ of size $m \times n$ is an array of numbers:

A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}

2. Basic Operations

2.1 Matrix Addition

For matrices $A$ and $B$ of the same size:

(A + B)_{ij} = A_{ij} + B_{ij}

2.2 Scalar Multiplication

For scalar $c$ and matrix $A$:

(cA)_{ij} = c \cdot A_{ij}

2.3 Matrix Multiplication

For $A$ of size $m \times n$ and $B$ of size $n \times p$:

(AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}

Note: Matrix multiplication is not commutative: $AB \neq BA$ in general.

3. Special Matrices

Identity Matrix: $I_n = \text{diag}(1, 1, \ldots, 1)$
Zero Matrix: $0_{m \times n}$ with all entries zero
Diagonal Matrix: $D = \text{diag}(d_1, d_2, \ldots, d_n)$
Symmetric Matrix: $A^T = A$
Skew-Symmetric: $A^T = -A$

4. Transpose

The transpose of matrix $A$ is:

(A^T)_{ij} = A_{ji}

Properties:

$(A^T)^T = A$
$(A + B)^T = A^T + B^T$
$(AB)^T = B^T A^T$
$(cA)^T = cA^T$

5. Determinant

For a $2 \times 2$ matrix:

\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

For larger matrices, use cofactor expansion or row operations.

6. Inverse

A matrix $A$ is invertible if $\det(A) \neq 0$. The inverse satisfies:

AA^{-1} = A^{-1}A = I

For $2 \times 2$ matrix:

A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

7. Rank

The rank of a matrix is the maximum number of linearly independent rows (or columns).

$\text{rank}(A) \leq \min(m, n)$
$\text{rank}(A) = \text{rank}(A^T)$
$\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))$

8. Row Operations

Elementary row operations:

Row Swap: Exchange two rows
Row Scaling: Multiply a row by a non-zero scalar
Row Addition: Add a multiple of one row to another

9. Row Echelon Form (REF)

A matrix is in REF if:

All zero rows are at the bottom
Leading coefficient (pivot) is 1
Pivots are to the right of pivots in rows above

10. Reduced Row Echelon Form (RREF)

In addition to REF properties:

All entries above and below pivots are zero

11. Systems of Linear Equations

A system $Ax = b$ can be solved using:

Gaussian Elimination: Convert to REF
Gauss-Jordan: Convert to RREF
Matrix Inversion: $x = A^{-1}b$ (if $A$ is invertible)

12. Code Example

# Python code for matrix operations
import numpy as np

# Create matrices
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Basic operations
C = A + B  # Addition
D = A @ B  # Matrix multiplication
E = A.T    # Transpose

# Determinant and inverse
det_A = np.linalg.det(A)
inv_A = np.linalg.inv(A)

# Solve system Ax = b
b = np.array([1, 2])
x = np.linalg.solve(A, b)

print(f"Determinant: {det_A}")
print(f"Inverse:\n{inv_A}")
print(f"Solution: {x}")

13. References

Strang, G. (2016). Introduction to Linear Algebra.
Lay, D. C. (2015). Linear Algebra and Its Applications.
Hoffman, K., & Kunze, R. (1971). Linear Algebra.