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Matrix Multiplication & Power Visualization

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Advanced Matrix Operations Explained

I. Matrix Multiplication ($A \times B$)

Matrix multiplication is not simply multiplying corresponding elements. If matrix $A$ is an $m \times n$ matrix and matrix $B$ is an $n \times p$ matrix, then their product $C = A \times B$ will be an $m \times p$ matrix.

Note: The prerequisite for matrix multiplication is that the number of columns in the first matrix must equal the number of rows in the second matrix.

Element $c_{ij}$ at row $i$, column $j$ of product matrix $C$ equals the dot product of row $i$ of matrix $A$ and column $j$ of matrix $B$:

$$c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

Important Properties of Matrix Multiplication:

  • Not commutative: In general $A \times B \neq B \times A$ (even if both products are defined).
  • Associative: $(A \times B) \times C = A \times (B \times C)$.
  • Distributive: $A \times (B + C) = A \times B + A \times C$.

II. Matrix Power ($A^k$)

When a matrix is a square matrix (number of rows equals number of columns, i.e., $n \times n$), we can raise it to a power. The $k$-th power of a square matrix is the matrix multiplied by itself $k$ times.

Let $A$ be an $n \times n$ square matrix, and $k$ be a positive integer:

$$A^k = \underbrace{A \times A \times \cdots \times A}_{k \text{ times } A}$$

Specifically, the 0-th power of a non-zero square matrix is defined as the identity matrix of the same order: $A^0 = I$

Matrix powers have important applications in computer graphics, Markov chain state transitions, and graph theory. For example, in graph theory, $A^k$ (adjacency matrix raised to power $k$) represents the number of paths of length $k$ between vertices.