## 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]```