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.