Eigen Values, Eigen Vectors

Note:

$ λ_1 $ x $λ_2$ x…x $λ_n$ = |A|

If $ λ_1, λ_2, …, λ_n $ are eigen values of A, then $ 1/λ_1, 1/λ_2, …, 1/λ_n $ are eigen values of $A^{-1}$

Eigen values of A and A’ are same.

$A^{-1}$ exists iff $λ_j$ ≠ 0, j = 1,2,.., n.

Orthogonal eigen vectors:

Two eigen vectors $X_1$ & $X_2$ are said to be orthogonal if $X’_1 X_2 = 0$

Note: Eigen values may be zero; an eigen vector may not be the zero vector.

For 2 x 2 matrix,
$λ^2 – S_1 λ + |A| = 0$
$S_1$ = Sum of minors of order one along main diagonal of A

For 3 x 3 matrix,
$λ^3 – S_1 λ^2 + S_2 λ – |A| = 0$
$S_1$ = Sum of minors of order one along main diagonal of A
$S_1$ = $a_{11} + a_{22} + a_{33}$

$S_2$ = Sum of minors of order two of the diagonal element of A
$S_2$ = minor of $a_{11}$ + minor of $a_{22}$ + minor of $a_{33}$

To check if eigen values are correct:

(1) For 2×2 matrix,
$λ_1 + λ_2 = a_{11} + a_{22}$ and $λ_1 λ_2 = |A|$

(2) For 3×3 matrix,

$λ_1 + λ_2 + λ_3 = a_{11} + a_{22} + a_{33}$ and $λ_1 λ_2 λ_3 = |A|$

Camer’s Rule:

Consider, $a_1x+b_1y+c_1z = 0$
$a_2x+b_2y+c_2z = 0$

Then,
${x}/{|\table b_1,c_1;b_2,c_2|} = {-y}/{|\table a_1,c_1;a_2,c_2|} = {z}/{|\table a_1,b_1;a_2,b_2|}$