Orthogonal & Orthonormal Vectors

Consider 2 column matrices and :

Product

The product of and is defined as:

Orthogonal

Orthogonal matrix

For a square matrix with real entries, it is orthogonal iff

Orthogonal vectors

and are orthogonal iff .

For a set of column vectors, they are orthogonal iff they are pairwise orthogonal. That is:

Note

are orthogonal are linearly independent.

Converse is not true.

Orthonormal

For a set of column vectors, they are orthonormal iff:

• They are pairwise orthogonal AND
• For all column vectors their norm is
  

Properties of orthogonal matrices

• Product of 2 orthogonal matrices of the same order is also an orthogonal matrix
• The columns or rows of an orthogonal matrix form an orthogonal set of vectors