Unit 4: Matrix Theory

Special Types of Matrices

These are classifications based on how a matrix behaves when multiplied by itself.

1. Idempotent Matrix

A square matrix A is idempotent if A² = A.

Multiplying it by itself gives the same matrix back. The identity matrix (I) and the zero matrix (0) are simple examples.

2. Nilpotent Matrix

A square matrix A is nilpotent if Aᵏ = 0 (the zero matrix) for some positive integer k.

The smallest such `k` is called the index of nilpotency.

3. Involutory Matrix

A square matrix A is involutory if A² = I (the identity matrix).

An involutory matrix is its own inverse (A = A⁻¹). The identity matrix (I) is a simple example. A reflection matrix is also involutory.

Matrix Operations (Transpose and Conjugate)

1. Transpose of a Matrix (Aᵀ)

The transpose of a matrix A, denoted Aᵀ, is the matrix obtained by interchanging its rows and columns.

If A is `m × n`, then Aᵀ is `n × m`. The element (Aᵀ)ᵢⱼ = Aⱼᵢ.

2. Conjugate of a Matrix (Ā)

The conjugate of a matrix A, denoted Ā, is the matrix obtained by replacing each element with its complex conjugate. (e.g., `a + bi` becomes `a - bi`).

If a matrix contains only real numbers, then Ā = A.

3. Transpose-Conjugate (A* or Aᴴ)

The transpose-conjugate (or Hermitian conjugate) of A is the transpose of the conjugate: A* = (Ā)ᵀ.

This operation is crucial for complex matrices, similar to what the transpose is for real matrices.

Symmetric and Hermitian Matrices

These matrices describe different types of "symmetry" in square matrices.

1. Symmetric Matrix (for Real Matrices)

A square matrix A is symmetric if it is equal to its transpose: Aᵀ = A.

This means `aᵢⱼ = aⱼᵢ` for all i, j. The matrix is a "mirror image" across its main diagonal.

2. Skew-Symmetric Matrix (for Real Matrices)

A square matrix A is skew-symmetric if it is equal to the negative of its transpose: Aᵀ = -A.

This means `aᵢⱼ = -aⱼᵢ`. This implies that all main diagonal elements must be zero (since `aᵢᵢ = -aᵢᵢ` means `2aᵢᵢ = 0`).

3. Hermitian Matrix (for Complex Matrices)

A square matrix A is Hermitian if it is equal to its transpose-conjugate: A* = A.

This means `aᵢⱼ = āⱼᵢ`. This implies that all main diagonal elements must be real numbers (since `aᵢᵢ = āᵢᵢ`).

4. Skew-Hermitian Matrix (for Complex Matrices)

A square matrix A is skew-Hermitian if it is equal to the negative of its transpose-conjugate: A* = -A.

This means `aᵢⱼ = -āⱼᵢ`. This implies that all main diagonal elements must be purely imaginary or zero (since `aᵢᵢ = -āᵢᵢ`).

Adjoint of a Square Matrix

Cofactor

First, the cofactor Cᵢⱼ of an element aᵢⱼ is the determinant of the submatrix obtained by deleting row `i` and column `j`, multiplied by `(-1)ⁱ⁺ʲ`.

Adjoint Matrix

The adjoint of a square matrix A, denoted adj(A), is the transpose of its cofactor matrix.

Steps to find adj(A):

1. Create the "matrix of minors" by finding the determinant of the submatrix for each element.
2. Create the "matrix of cofactors" (C) by applying the sign rule `(-1)ⁱ⁺ʲ` to each minor.
3. The adjoint is the transpose of this cofactor matrix: adj(A) = Cᵀ.

Inverse of a Square Matrix

A square matrix A is invertible (or non-singular) if there exists a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I.

Test for Invertibility

A matrix A is invertible if and only if its determinant is non-zero: det(A) ≠ 0.

Formula for the Inverse

The inverse of an invertible matrix A is given by the formula:

A⁻¹ = (1 / det(A)) * adj(A)

This formula connects the inverse, determinant, and adjoint, and is a fundamental property of matrices.

Jacobi's Theorem

This theorem provides a direct relationship between the determinant of a matrix and the determinant of its adjoint. It is derived from the property A · adj(A) = det(A) · I.

Jacobi's Theorem states:
If A is an invertible square matrix of order n, then the determinant of its adjoint is:
det(adj(A)) = (det(A))ⁿ⁻¹