Orthogonal vs Orthonormal
In mathematics, the two words orthogonal and orthonormal are frequently used along with a set of vectors. Here, the term ‘vector’ is used in the sense that it is an element of a vector space – an algebraic structure used in linear algebra. For our discussion, we will consider an inner-product space – a vector space V along with an inner product [ ] defined on V.
As an example, for an inner product, space is the set of all 3-dimensional position vectors along with the usual dot product.
What is orthogonal?
A nonempty subset S of an inner product space V is said to be orthogonal, if and only if for each distinct u, v in S, [u, v] = 0; i.e. the inner product of u and v is equal to the zero scalar in the inner product space.
For example, in the set of all 3-dimensional position vectors, this is equivalent to saying that, for each distinct pair of position vectors p and q in S, p and q are perpendicular to each other. (Remember that the inner product in this vector space is the dot product. Also, the dot product of two vectors is equal to 0 if and only if the two vectors are perpendicular to each other.)
Consider the set S = {(0,2,0), (4,0,0), (0,0,5)}, which is a subset of the 3-dimensional position vectors. Observe that (0,2,0).(4,0,0) = 0, (4,0,0).(0,0,5) = 0 & (0,2,0).(0,0,5) = 0. Hence, the set S is orthogonal. In particular, two vectors are said to be orthogonal if their inner product is 0. Therefore, each pair of vectors in Sis orthogonal.
What is orthonormal?
A nonempty subset S of an inner product space V is said to be orthonormal if and only if S is orthogonal and for each vector u in S, [u, u] = 1. Therefore, it can be seen that every orthonormal set is orthogonal but not vice versa.
For example, in the set of all 3-dimensional position vectors, this is equivalent to saying that, for each distinct pair of position vectors p and q in S, p and q are perpendicular to each other, and for each p in S, |p| = 1. This is because the condition [p, p] = 1 reduces to p.p=|p||p|cos0 = |p|2=1, which is equivalent to |p| = 1. Therefore, given an orthogonal set we can always form a corresponding orthonormal set by dividing each vector by its magnitude.
T = {(0,1,0), (1,0,0), (0,0,1)} is an orthonormal subset of the set of all 3-dimensional position vectors. It is easy to see that it was obtained by dividing each of the vectors in the set S, by their magnitudes.
What is the difference between orthogonal and orthonormal?
- A nonempty subset S of an inner product space V is said to be orthogonal, if and only if for each distinct u, v in S, [u, v] = 0. However, it is orthonormal, if and only if an additional condition – for each vector u in S, [u, u] = 1 is satisfied.
- Any orthonormal set is orthogonal but not vice-versa.
- Any orthogonal set corresponds to a unique orthonormal set but an orthonormal set may correspond to many orthogonal sets.