Matrix Theory: Definitions and Operations

Definition

• A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Notation

• A matrix is typically denoted by uppercase letters (e.g., A, B, C).
• The element in the i-th row and j-th column is denoted as ( a_{ij} ).

Types of Matrices

1. Row Matrix: A matrix with only one row (1 x n).
2. Column Matrix: A matrix with only one column (m x 1).
3. Square Matrix: A matrix with the same number of rows and columns (n x n).
4. Zero Matrix: A matrix where all elements are zero.
5. Diagonal Matrix: A square matrix where all off-diagonal elements are zero.
6. Identity Matrix: A diagonal matrix where all diagonal elements are 1.
7. Transpose Matrix: The matrix obtained by swapping rows and columns (denoted as ( A^T )).

Operations

1. Addition:

   • Matrices can be added if they have the same dimensions.
   • ( (A + B){ij} = a{ij} + b_{ij} ).

2. Subtraction:

   • Similar to addition, matrices must have the same dimensions.
   • ( (A - B){ij} = a{ij} - b_{ij} ).

3. Scalar Multiplication:

   • Each element of the matrix is multiplied by a scalar ( k ).
   • ( (kA){ij} = k \cdot a{ij} ).

4. Matrix Multiplication:

   • The number of columns in the first matrix must equal the number of rows in the second.
   • ( (AB){ij} = \sum{k} a_{ik} b_{kj} ).

Determinants

• A scalar value that can be computed from the elements of a square matrix.
• Useful for solving linear equations, analyzing invertibility, and understanding geometric properties.

Inverses

• The inverse of a matrix ( A ) is denoted as ( A^{-1} ).
• A matrix is invertible if ( AB = I ) where ( I ) is the identity matrix.
• The inverse exists only for square matrices with a non-zero determinant.

Eigenvalues and Eigenvectors

• Eigenvalue (λ): A scalar such that there exists a non-zero vector ( v ) (eigenvector) where ( Av = λv ).
• Useful in various applications including stability analysis, quantum mechanics, and computer graphics.

Applications

• Used in solving systems of linear equations.
• Foundational in computer science, economics, engineering, physics, and statistics.

Key Properties

• Commutative Property: Addition is commutative (A + B = B + A), but multiplication is not (AB ≠ BA in general).
• Associative Property: (A + B) + C = A + (B + C) and (AB)C = A(BC).
• Distributive Property: A(B + C) = AB + AC.

Special Cases

• Singular Matrix: A square matrix that does not have an inverse (determinant = 0).
• Orthogonal Matrix: A square matrix whose rows and columns are orthogonal unit vectors (i.e., ( A^TA = I )).

Definition

• A matrix is a rectangular array of numbers or symbols arranged in rows and columns.

Notation

• Uppercase letters represent matrices (e.g., A, B, C).
• The element at the i-th row and j-th column is denoted as ( a_{ij} ).

Types of Matrices

• Row Matrix: Contains a single row (1 x n).
• Column Matrix: Contains a single column (m x 1).
• Square Matrix: Equal number of rows and columns (n x n).
• Zero Matrix: All elements are zero.
• Diagonal Matrix: Only diagonal elements are non-zero with all off-diagonal elements zero.
• Identity Matrix: Diagonal matrix with all diagonal elements equal to 1.
• Transpose Matrix: Rows and columns are swapped, denoted as ( A^T ).

Operations

• Addition: Can be performed between matrices of the same dimensions, calculated as ( (A + B){ij} = a{ij} + b_{ij} ).
• Subtraction: Similar to addition for matrices of the same dimensions, calculated as ( (A - B){ij} = a{ij} - b_{ij} ).
• Scalar Multiplication: Each element is multiplied by a scalar ( k ) as ( (kA){ij} = k \cdot a{ij} ).
• Matrix Multiplication: Requires that the number of columns in the first matrix equals the number of rows in the second, calculated as ( (AB){ij} = \sum{k} a_{ik} b_{kj} ).

Determinants

• A scalar value derived from a square matrix's elements, essential for solving linear equations, checking invertibility, and understanding geometric properties.

Inverses

• The inverse of a matrix ( A ) is represented as ( A^{-1} ).
• A matrix is invertible if ( AB = I ), where ( I ) is the identity matrix.
• Only square matrices with a non-zero determinant possess an inverse.

Eigenvalues and Eigenvectors

• Eigenvalue (λ): A scalar related to a non-zero vector ( v ) (eigenvector) fulfilling the equation ( Av = λv ).
• Significant in various fields such as stability analysis, quantum mechanics, and computer graphics.

Applications

• Widely used for solving systems of linear equations.
• Fundamental in disciplines like computer science, economics, engineering, physics, and statistics.

Key Properties

• Commutative Property: Addition is commutative (A + B = B + A), multiplication is not (AB ≠ BA generally).
• Associative Property: ( (A + B) + C = A + (B + C) ), ( (AB)C = A(BC) ).
• Distributive Property: ( A(B + C) = AB + AC ).

Special Cases

• Singular Matrix: Square matrix lacking an inverse (determinant = 0).
• Orthogonal Matrix: Square matrix with orthogonal unit vector rows and columns, satisfying ( A^TA = I ).