Symbolic Math Toolbox

Linear Algebra

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 simple function

• A = simple(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 = simple(I)
  I =
  [1, 0]
  [0, 1]
  

Another Example

Linear Algebraic Operations