PROPERTIES OF EIGEN VALUES :-






 PROPERTIES OF EIGEN VALUES:-

1.     The Sum of the Eigen values of a square matrix is equal to its Trace and Product of the Eigen values is equal to its Determinant.
That is, A is an nxn matrix and l1, l2, l3, l4 are Eigen values then
l1+l2+l3+l4=Trace A
l1.l2.l3.l4=
2.     If ‘l’ is an Eigen value of A corresponding to Eigen vector ‘X’ then ln is Eigen value of An corresponding to Eigen vector ‘X’.
3.     A square matrix ‘A’ and its transpose “AT” have the same Eigen values.
4.     If A and B are n-rowed square matrices and “A” is invertible then A-1B and BA-1 have the same Eigen values.
5.     If l1, l2, ---, ln are the Eigen values of a matrix “A” then kl1, kl2, ---, kln are the Eigen values of a matrix “KA”.  Where K is non-zero scalar.
6.     If ‘l’ is an Eigen value of Matrix “A” then l+k” is an Eigen value of matrix “A+KI”, K is non-zero scalar.
7.     If l1, l2, ---, ln are the Eigen values of a matrix “A” then l1-K, l2-K, ---, ln-K are Eigen values of matrix A-KI.
8.     If l1, l2, ---, ln are the Eigen values of a matrix “A” then the Eigen values of matrix          (A-lI)2= (l1-l)2, (l2-l)2,….,(ln-l)2.
9.     If l us an Eigen value of a non-singular matrix ‘A’ corresponding to the Eigen          vector ‘X’ then l-1 is an Eigen value of A-1 and corresponding Eigen 
            vector ‘X’ itself.
10.            If ‘l’ is an Eigen value of a non-singular matrix ‘A’ then  is an Eigen value 
               of matrix ‘Adj A’.
11.            If ‘l’ is an Eigen value of an Orthogonal matrix then  is also an Eigen value.
12.            If ‘l’ is an Eigen value of A then the Eigen value of B=a0A2+a1A+a2I is                      a0l2+a1l+a2.
13.            If A and P be square matrices of order ‘n’ such that P is non-singular then A &       P-1AP have the same Eigen values.
14.            If A and B are non-singular matrices of the same order then AB and BA have the        same Eigen values.
15.            The Eigen values of a “Triangular Matrix” are just the diagonal elements of the           matrix
16.            The Eigen values of a Diagonal Matrix are just the diagonal elements of the                 matrix.
17.            The Eigen values of a real symmetric matrix are always real.
18.            For a real symmetric matrix, the Eigen vectors corresponding to two distinct              Eigen values are orthogonal.

19.            The two Eigen vectors corresponding to the two different Eigen values are      
                Linearly Independent.

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