Orthogonal

Consider 2 column matrices and :

Product

The product of and is defined as:

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.

Orthogonal matrix

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

A matrix is orthogonal iff sum of the squared elements of any row or column is .

Properties

• is invertible, non-singular
• are orthogonal
• It is diagonalizable over (may not be, over )
• 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