Eigen Values or Characteristic roots and Characteristic Vectors of a square matrix.
Let X be any non-zero vector satisfying `AX = lambdaX`
where `lambda` is any scalar, then `lambda` is said to be eigenvalue OJ: characteristic root of square matrix A and the vector X is called eigen vector or characteristic vector of matrix A.
Now, from Eq. (i), we have `(A- lamdaI) X= O`
Since, `x != 0,` we deduce that the matrix `(A-lambdaI)` is singular, so that its determinant is 0 .
`| A - λI | = 0`
is called characteristic equation of matrix A.
If A be `n xx n` matrix, then equation `| A - λI | = 0` reduces to polynomial equation of nth from degree in `λ` which given n values of ` λ` i.e., matrix A will have n characteristic roots or eigen values.
`text( Properties of Eigen Values :)`
(i) Any square matrix A and its transpose `A^T` have the same eigen values.
(ii) The sum of the eigen values of a matrix is equal to the trace of the matrix.
(iii) The product the eigen values of a matrix A is equal to the determinant of A.
(iv) If `λ_1, λ_2, λ_3, ........λ_n`are the eigenvalues of A, then the eigenvalues of
a) kA are `kλ_1, kλ_2, kλ_3, ........kλ_n`
b) `A^m` are `(λ_1)^m, (λ_2)^m, (λ_3)^m, ........(λ_n)^m`
c) `A^(-1)` are `1/λ_1, 1/λ_2, 1/λ_3, ........1/λ_n`
`text(Note :)`
1.. All the eigen values of a real symmetric matrix are real and the eigen vectors corresponding to two distinct eigen values are orthogonal.
2.All the eigen values of a real skew-symmetric matrix are purel imaginary or zero. An odd order skew-symmetric matrix is singular and hence has zero as an eigen value.
Eigen Values or Characteristic roots and Characteristic Vectors of a square matrix.
Let X be any non-zero vector satisfying `AX = lambdaX`
where `lambda` is any scalar, then `lambda` is said to be eigenvalue OJ: characteristic root of square matrix A and the vector X is called eigen vector or characteristic vector of matrix A.
Now, from Eq. (i), we have `(A- lamdaI) X= O`
Since, `x != 0,` we deduce that the matrix `(A-lambdaI)` is singular, so that its determinant is 0 .
`| A - λI | = 0`
is called characteristic equation of matrix A.
If A be `n xx n` matrix, then equation `| A - λI | = 0` reduces to polynomial equation of nth from degree in `λ` which given n values of ` λ` i.e., matrix A will have n characteristic roots or eigen values.
`text( Properties of Eigen Values :)`
(i) Any square matrix A and its transpose `A^T` have the same eigen values.
(ii) The sum of the eigen values of a matrix is equal to the trace of the matrix.
(iii) The product the eigen values of a matrix A is equal to the determinant of A.
(iv) If `λ_1, λ_2, λ_3, ........λ_n`are the eigenvalues of A, then the eigenvalues of
a) kA are `kλ_1, kλ_2, kλ_3, ........kλ_n`
b) `A^m` are `(λ_1)^m, (λ_2)^m, (λ_3)^m, ........(λ_n)^m`
c) `A^(-1)` are `1/λ_1, 1/λ_2, 1/λ_3, ........1/λ_n`
`text(Note :)`
1.. All the eigen values of a real symmetric matrix are real and the eigen vectors corresponding to two distinct eigen values are orthogonal.
2.All the eigen values of a real skew-symmetric matrix are purel imaginary or zero. An odd order skew-symmetric matrix is singular and hence has zero as an eigen value.