Matrix Operations
Matrix Multiplication
For matrices \(A \in \mathbb{R}^{m \times n}\) and \(B \in \mathbb{R}^{n \times p}\), the product \(C = AB\) is defined as:
\[C_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}\]
Properties
- Associativity: \((AB)C = A(BC)\)
- Distributivity: \(A(B + C) = AB + AC\)
- Non-commutativity: Generally \(AB \neq BA\)
Eigenvalues and Eigenvectors
For a square matrix \(A \in \mathbb{R}^{n \times n}\), a scalar \(\lambda\) is an eigenvalue if there exists a non-zero vector \(\mathbf{v}\) such that:
\[A\mathbf{v} = \lambda\mathbf{v}\]
The vector \(\mathbf{v}\) is called an eigenvector corresponding to eigenvalue \(\lambda\).
Characteristic Polynomial
Eigenvalues are found by solving:
\[\det(A - \lambda I) = 0\]
Example: 2×2 Matrix
Consider the matrix: \[A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}\]
The characteristic polynomial is: \[\det\begin{pmatrix} 3-\lambda & 1 \\ 0 & 2-\lambda \end{pmatrix} = (3-\lambda)(2-\lambda) = 0\]
Therefore, eigenvalues are \(\lambda_1 = 3\) and \(\lambda_2 = 2\).
Applications in Network Theory
In telecommunications, eigenvalue decomposition is used for:
- Channel capacity analysis in MIMO systems
- Network stability analysis using adjacency matrices
- Signal processing for beamforming algorithms
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