This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

Two non-zero vectors $v$ and $w$ in an inner product space are orthogonal if $\langle v, w\rangle = 0$. Note that $v$ and $w$ are orthogonl if and only if the line spanned by $v$ and the line spanned by $w$ are perpendicular (the angle between them is $90^{\circ}$ or $\frac{\pi}{2}$ radians).

A set of non-zero vectors is called (pairwise) orthogonal if each pair of vectors in the set are orthogonal. Note that if a set of non-zero vectors is orthogonal, then it is linearly independent. If in addition to being orthogonal, all the vectors in the set have length one, it is called an orthonormal set.

A square matrix $A$ is called orthogonal if it is invertible and $A^T = A^{-1}$. A matrix $A$ is orthogonal if and only if the rows of $A$ are orthonormal and the columns of $A$ are orthonormal.

A linear transformation $T$ from an inner product space $V$ to itself is called orthogonal if $\langle T(v), T(w)\rangle = \langle v, w\rangle$ for all $v, w \in V$. Note that if $V$ is finite-dimensional, and we fix a basis for $V$, then $T(v) = Av$ for some square matrix $A$. In that case, $T$ is orthogonal if and only if $A$ is an orthogonal matrix.