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Questions and Answers
What is the purpose of finding the characteristic equation of a square matrix?
What is the name of the theorem that states every square matrix satisfies its own characteristic equation?
What is the condition for the system of equations AX = λX to have a non-trivial solution?
What is the name given to the non-zero vectors X that satisfy the equation AX = λX?
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What is the formula for the characteristic equation of a 3x3 matrix A?
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What is the process of finding a diagonal matrix B such that B = A'BA?
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What is the name given to the matrix B in the equation B = A'BA?
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What is the purpose of finding the eigen values of a matrix?
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What is the equation that is used to find the eigen values of a matrix?
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What is the condition for a matrix A to be diagonalised?
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Study Notes
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.
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Description
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!
