Orthogonal

Definition

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

Condition

A is a 3 by 3 matrix.
$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix}$
A is an orthogonal matrix if inner product values of two columns are 0. In other words, $ab + de + gh = 0$ and $bc + ef + hi = 0$ and $ca + fd + ig = 0$.

Unit vector

Definition

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

Condition

A is a unit vector if inner product values of a column are 1. In other words, $a^2 + d^2 + g^2 = 1$ and $b^2 + e^2 + h^2 = 1$ and $c^2 + f^2 + i^2 = 1$.

Orthonormal

Definition

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

Condition

A is an orthonormal matrix if A is an orthogonal matrix and a unit vector.