A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental structures in linear algebra that can represent linear transformations, systems of equations, and data structures.
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Matrix Representation
An m × n matrix A has m rows and n columns:
Matrix Operations
Addition and Subtraction
For matrices of the same size:
Scalar Multiplication
For a scalar c:
Matrix Multiplication
For matrices A (m × n) and B (n × p):
Special Matrices
Square Matrix: Number of rows equals number of columns
Identity Matrix: Square matrix with 1s on diagonal, 0s elsewhere
Diagonal Matrix: Non-zero elements only on diagonal
Symmetric Matrix: Equal to its transpose (A = Aᵀ)
Triangular Matrix: Upper or lower triangular form
Properties
Determinant: A scalar value that provides information about the matrix's invertibility and scaling properties
Rank: The dimension of the vector space spanned by the matrix's columns
Trace: Sum of diagonal elements in a square matrix
Inverse: For a square matrix A, matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I
Applications
Matrices are used extensively in:
Solving systems of linear equations
Computer graphics (transformations)
Machine learning (data representation)
Quantum mechanics
Network analysis
Economics (input-output models)
Computational Considerations
The time complexity for common matrix operations:
Matrix Addition: O(mn) for m × n matrices
Matrix Multiplication: O(mnp) for m × n and n × p matrices
Matrix Determinant: O(n³) for n × n matrices
Matrix Inverse: O(n³) for n × n matrices
Space complexity is typically O(mn) for an m × n matrix.