Application of Eigen Value Problem: Stretching of an Elastic Membrane

APPLICATION OF EIGEN VALUE PROBLEM:

STRETCHING OF AN ELASTIC MEMBRANE

If is a vector and m is real number, then is a vector having the same direction as if m> 0 and the opposite direction if m < 0.

If m = 0, then

Geometrically, stretches if m > 1, contacts if 0 < m < 1 and reverses if m < 0.

If then we write (x1, x2) which is the position vector of the point P(x1, x2).

3 = (3x1, 3x2) is the position vector of the point Q(3x1, 3x2). Here 3 stretches in the same direction.

Example 1

Thus Y = AX, which is a linear transformation.

We can regard X as the position vector of the point P(x1, x2) and Y as the position vector of the point Q(y1, y2) in the xy-plane.

The transformation Y = AX carries the point P into the point Q and

For example,

(i) If X = (x1, x2) = (1,1) then y1 = 5 × 1 + 3 × 1 = 8 and y2 = 3 × 1 + 5 × 1 = 8

⸫ Y = (y1, y2) = (8,8) = 8(1,1) = 8X

Hence X is stretched to Y in the same direction

(ii) If X= (1, −1), then y1 = 5 × 1 + 3 × (-1) = 2

and y2 = 3 × 1 + 5 × (−1) = −2

⸫ Y = (y1, y2) = (2, -2) = 2(1, -1) = 2X

Hence X is stretched to Y in the same direction.

(iii) If X = (2, 1), then y1 = 5.2 + 3.1 = 13

y2 = 3.2 + 5.1 = 11

⸫ Y = (y1, y2) = (13, 11)

So, Y ≠ m X for any real m

Hence directions of X and Y are different.

Note 

Suppose X = (1, 1) is stretched to Y = (-2, −2) by some transformation, then Y = -2 (1, 1) = -2X

⸫ X is stretched to Y in the opposite direction of X.

Example 2

Consider an elastic membrane in the xy-plane. If a point P(x1, x2) lying on the boundary circle x2 + y2 = 1 is stretched to the point Q(y1, y2) by the transformation Y = AX, where A = then find the curve on which Q lies.

Solution

Solving (1) and (2) for x1 and x2, we get

which represents an ellipse

Note 

The general second degree equation ax2 + 2hxy + by2 + 2gx +2fy + c = 0 will represent an ellipse if h2 – ab < 0, a hyperbola if h2 – ab > 0, a parabola if h2 – ab = 0, and a circle if a = b and h = 0.

In the equation (4) a = 34, h = -30, b = 34

⸫ h2 – ab = (−30)2 – 34.34 = − 256 < 0

If the center of the ellipse is chosen as the origin and the major and the minor axis of the ellipse are chosen as the coordinate axes, then the ellipse reduces to canonical form or standard form

1. Principal Directions

Let P be a point in an elastic membarane in the xy plane with position vector X. Let P be pushed to a point Q in the plane with position vector Y by a linear transformation Y = AX, where A is 2 × 2 matrix. If the directions of X and Y are the same or opposite, then these directions are called Principal directions. 

In example 1, (i) Y= 8X and (ii) Y = 2X give principal directions, since the vector Y has the same direction as that of X.

Since Y = AX, we have AX= 8X and AX = 2X.

This means 8 and 2 are eigen values of A.

Hence principal directions are related to eigen values of A.