Some property of Eigen Values and Eigen vector
  1. The matrices A and AT have the same eigen values.                                                            we know that,
    (A-λI)=AT-λIT
    ∴ |(A-λI)T|=|AT-λI|                [i.e |AT|=|A|T]
    |A-λI|=|AT-λI|
    ∴ A & AT have same eigen values
  2. '0' is a characteristic root of a matrix if and only if the matrix is singular .i.e |A|=0
  3. The characteristic roots of a triangular matrix are just the diagonal elements of the matrix.
  4. If λ1,λ2...........λn are the eigen values of A ,then  kλ1,kλ2,kλ3......... are eigen values of kA.
  5. If  'A' is non singular, then eigen values of A-1 are the reciprocals of eigen values of 'A'.
  6. If 'λ' is a charactristicroot of matrix 'A' then (k+λ) is a characteristic root of matrix (A+kI).
  7. |A|=\prod ^{n}_{i=1}\lambda _{i}
  8. trace(A)=\sumλi  ,(i=1 to i=n)
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