Square Matrix Decomposition into Symmetric and Skew-Symmetric Matrices
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Questions and Answers

What is the purpose of finding the characteristic equation of a square matrix?

  • To find the inverse of the matrix
  • To find the rank of the matrix
  • To find the determinant of the matrix
  • To find the eigen values of the matrix (correct)
  • What is the name of the theorem that states every square matrix satisfies its own characteristic equation?

  • Matrix Inverse Theorem
  • Cayley-Hamilton Theorem (correct)
  • Eigen Value Theorem
  • Determinant Theorem
  • What is the condition for the system of equations AX = λX to have a non-trivial solution?

  • |A - λI| = 0 (correct)
  • Det(A) = 0
  • |A - λI| ≠ 0
  • Det(A) ≠ 0
  • What is the name given to the non-zero vectors X that satisfy the equation AX = λX?

    <p>Characteristic Vectors</p> Signup and view all the answers

    What is the formula for the characteristic equation of a 3x3 matrix A?

    <p>λ^3 + a₁λ^2 + a₂λ + a₃ = 0</p> Signup and view all the answers

    What is the process of finding a diagonal matrix B such that B = A'BA?

    <p>Diagonalisation of Matrices</p> Signup and view all the answers

    What is the name given to the matrix B in the equation B = A'BA?

    <p>Modal Matrix</p> Signup and view all the answers

    What is the purpose of finding the eigen values of a matrix?

    <p>To diagonalise the matrix</p> Signup and view all the answers

    What is the equation that is used to find the eigen values of a matrix?

    <p>(A - λI)X = 0</p> Signup and view all the answers

    What is the condition for a matrix A to be diagonalised?

    <p>A must have distinct eigen values</p> Signup and view all the answers

    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!

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