Calculating Eigenvalues in Linear Algebra

Calculating Eigen Values

Definition

• An eigenvalue is a scalar that represents how much a linear transformation changes a vector.
• It is a scalar that satisfies the equation Ax = λx, where A is a square matrix, x is a non-zero vector, and λ is the eigenvalue.

Methods for Calculating Eigen Values

• Characteristic Equation
  • |A - λI| = 0, where A is the square matrix, λ is the eigenvalue, and I is the identity matrix.
  • This equation is used to find the eigenvalues of a matrix.
• Determinant Method
  • det(A - λI) = 0, where det() is the determinant of the matrix.
  • This method is used to find the eigenvalues of a matrix.
• Factoring Method
  • Factor the characteristic polynomial to find the eigenvalues.
  • This method is used to find the eigenvalues of a matrix.

Example

• Find the eigenvalues of the matrix A = | 2 1 | | 2 3 |

• Calculate the characteristic equation: |A - λI| = 0

• |A - λI| = | 2-λ 1 | | 2 3-λ |

• Simplify the equation: λ^2 - 5λ + 4 = 0

• Factor the equation: (λ - 4)(λ - 1) = 0

• The eigenvalues are λ = 4 and λ = 1.

Important Properties

• The eigenvalues of a matrix are the roots of the characteristic equation.
• The eigenvalues of a matrix are scalar values.
• The eigenvalues of a matrix can be real or complex numbers.

Calculating Eigen Values

Definition of Eigen Values

• An eigenvalue is a scalar that represents how much a linear transformation changes a vector.
• It is a scalar that satisfies the equation Ax = λx, where A is a square matrix, x is a non-zero vector, and λ is the eigenvalue.

Methods for Calculating Eigen Values

Characteristic Equation Method

• The characteristic equation is |A - λI| = 0, where A is the square matrix, λ is the eigenvalue, and I is the identity matrix.
• This equation is used to find the eigenvalues of a matrix.

Determinant Method

• The determinant method involves finding the determinant of the matrix A - λI, which is set equal to 0.
• The equation is det(A - λI) = 0, where det() is the determinant of the matrix.

Factoring Method

• The factoring method involves factorizing the characteristic polynomial to find the eigenvalues.
• This method is used to find the eigenvalues of a matrix.

Example of Calculating Eigen Values

• To find the eigenvalues of a matrix, calculate the characteristic equation and simplify it to get a quadratic equation.
• Factor the equation to find the eigenvalues.

Important Properties of Eigen Values

• The eigenvalues of a matrix are the roots of the characteristic equation.
• The eigenvalues of a matrix are scalar values.
• The eigenvalues of a matrix can be real or complex numbers.

This quiz covers the definition and methods for calculating eigenvalues, including the characteristic equation in linear algebra.