Square Matrix Decomposition into Symmetric and Skew-Symmetric Matrices

Theorem 2: Uniqueness of Symmetric and Skew-Symmetric Matrices

• Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
• If A is a square matrix, then A = P + Q, where P is symmetric and Q is skew-symmetric.

Theorem 3: Uniqueness of Hermitian and Skew-Hermitian Matrices

• A square matrix can be uniquely expressed as the sum of a Hermitian matrix and a skew-Hermitian matrix.
• If A is a square matrix, then A = R + S, where R is Hermitian and S is skew-Hermitian.

Singular and Non-Singular Matrices

• A square matrix A is said to be singular if its determinant is zero.
• A square matrix A is said to be non-singular if its determinant is non-zero.

Inverse of a Matrix

• If two square matrices A and B are of the same order and AB = I, then A is called the inverse of B and vice versa.
• The inverse of A is denoted by A⁻¹.
• The following facts about inverses can be easily proven:
  • The inverse of a matrix is unique.
  • The inverse of a matrix exists if and only if A is non-singular.
  • If A and B are non-singular square matrices of the same order, then (AB)⁻¹ = B⁻¹A⁻¹.

Orthogonal Matrix

• A square matrix A is said to be orthogonal if AA' = I.
• If A is orthogonal, then A' is also orthogonal.
• The determinant of an orthogonal matrix is ±1.

Unitary Matrix

• A square matrix A is said to be unitary if AA' = I.
• If A is unitary, then A' is also unitary.

Theorem 5: Orthogonality of AB and BA

• If A and B are orthogonal matrices, then AB and BA are also orthogonal.

Theorem 6: Orthogonality of A and A'

• If A is an orthogonal matrix, then A' is also an orthogonal matrix.

Eigen Values and Eigen Vectors

• The characteristic value problem is to find the scalar λ and non-zero vectors X satisfying the equation AX = λX.
• The equation |A - λI| = 0 is called the characteristic equation.
• The roots of the equation |A - λI| = 0 are the eigenvalues or latent roots or characteristic values of A.
• The corresponding non-zero vectors X satisfying the equation (A - λI)X = 0 are called the eigenvectors or characteristic vectors of A.

Cayley-Hamilton Theorem

• Every square matrix satisfies its own characteristic equation.
• For a 3x3 matrix A, its characteristic equation is given by P(λ) = λ³ - a₁λ² + a₂λ - a₃ = 0, where a₁, a₂, and a₃ are constants.

Diagonalisation of Matrices

• Diagonalising a matrix A means finding another matrix B (called the modal matrix) such that B⁻¹AB is a diagonal matrix.
• The working rule to diagonalise a matrix A is to form the characteristic equation, find the eigenvalues, and then find the corresponding eigenvectors.

This quiz is about Theorem 2, which states that every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix. Learn how to prove this theorem!