Page 399 - Applied Numerical Methods Using MATLAB

388 MATRICES AND EIGENVALUES
First, let us consider a 2 × 2 matrix composed of two two-dimensional vectors
(2)
x (1) and x .

X = [ x (1) x (2) ] = x 11 x 12 (8.5.16)
x 21 x 22
Conclusively, the absolute value of the determinant of this matrix

det(X) =|X|= x 11 x 22 − x 12 x 21 (8.5.17)

equals the area of the parallelogram having the two vectors as its two neighboring
sides. In order to certify this fact, let us make a clockwise rotation of the two
vectors by the phase angle of x (1)

−θ 1 =− tan −1 x 21 (8.5.18)
x 11

so that the new vector y (1) corresponding to x (1) becomes aligned with the x 1 -axis
(see Fig. 8.2). For this purpose, we multiply our matrix X by the rotation matrix
defined by Eq. (8.4.2)

cos θ 1 − sin(−θ 1 ) 1 x 11 x 21
R(−θ 1 ) = = (8.5.19)
sin(−θ 1 ) cos θ 1 2 2 −x 21 x 11
x + x
11 21
to get

1 x 11 x 21 x 11 x 12
(8.5.20a)
Y = R(−θ 1 )X = −x 21 x 11 x 21 x 22
2
x + x 2
11 21
2
1 x + x 2 x 11 x 12 + x 21 x 22
[ y (1) y (2) ] = 11 0 21 (8.5.20b)
2
x + x 2 −x 12 x 21 + x 11 x 22
11 21
The parallelograms having the original vectors and the new vectors as their two
neighboring sides are depicted in Fig. 8.2, where the areas of the parallelograms
turn out to be equal to the absolute values of the determinants of the matrices X
and Y as follows:

Area of the parallelograms
= Length of the bottom side × Height of the parallelogram
(1)
(2)
= (x 1 component of y ) × (x 2 component of y ) = y 11 y 22 = det(Y)
2
x + x 2 −x 12 x 21 + x 11 x 22
11 21
× ≡ det(X) (8.5.21)
=
2
2
x + x 2 x + x 2
11 21 11 21