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Eigen values and Eigen Vector | Free Study Material For Mathematics | Linear Algebra | Matrix | Eigen values | Eigen Vector


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Eigen values and Eigen Vector

⚫Introduction
⚪ For a given square matrix A
► A - λI matrix is called Characteristic Matrix
► | A - λI| = 0 is Characteristic Equation
► The roots of | A - λI| = 0 are eigen values
Example :

Characteristic Equation :

Eigen Values or Characteristics Equation Roots :
λ3 - 7λ2 + 11λ - 5 = 0
(λ - 1)(λ-1)(λ - 5) =0
λ = 1 ,1 ,5
⚫ Important Properties Of Eigenvalues
► Any Square matrix A and its transpose A' have the same eigenvalue
► The sum of eigenvalues is equal to the trace of the matrix (sum of principal diagonal element)
► Product of eigenvalues is equal to the determinant of the matrix
► If λ1 , λ2 , λ3...... λn are the eigenvalues of A , then the Eigenvalues of
   ⚪ kA are kλ1 , kλ2 , kλ3...... kλn
   ⚪ Am are λ1m , λ2m , λ3m...... λnm
   ⚪ A-1 are 1/λ1 , 1/λ2 , 1/λ3 ........1/λn
► If λ is an eigenvalue of an orthogonal matrix , then 1/λ is also eigenvalue.
► The eigenvalues of a triangular martrix are equal to the diagonal elements of the matrix
► If A and B are two matrices of same order ,then the matrices AB and BA have the same eigenvalues.
► The maximum number of distinct eigenvalues is n where n is the size of the matrix A.
Practice set 1

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