Basic Algebraic Operations

Basic algebraic operations on symbolic objects are the same as operations on MATLAB^® objects of class double. This is illustrated in the following example.

The Givens transformation produces a plane rotation through the angle t. The statements

syms t
G = [cos(t) sin(t); -sin(t) cos(t)]

create this transformation matrix.

G =
[  cos(t),  sin(t)]
[ -sin(t),  cos(t)]

Applying the Givens transformation twice should simply be a rotation through twice the angle. The corresponding matrix can be computed by multiplying G by itself or by raising G to the second power. Both

A = G*G

and

A = G^2

produce

A =
[ cos(t)^2 - sin(t)^2,     2*cos(t)*sin(t)]
[    -2*cos(t)*sin(t), cos(t)^2 - sin(t)^2]

The simplify function

A = simplify(A)

uses a trigonometric identity to return the expected form by trying several different identities and picking the one that produces the shortest representation.

A =
[  cos(2*t),  sin(2*t)]
[ -sin(2*t),  cos(2*t)]

The Givens rotation is an orthogonal matrix, so its transpose is its inverse. Confirming this by

I = G.' *G

which produces

I =
[ cos(t)^2 + sin(t)^2,                   0]
[                   0, cos(t)^2 + sin(t)^2]

and then

I = simplify(I)

I =
[ 1, 0]
[ 0, 1]