Orthogonal family for a bilinear form

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B$ be a bilinear form on $V$. A family $(v_i)_{i \in I}$ of vectors of $V$ is orthogonal for $B$ when $$B(v_i, v_j) = 0 \qquad \text{for all } i, j \in I \text{ with } i \ne j.$$

Remarks

For an inner product, this is the usual notion of an orthogonal family. For a general $B$, the relation $B(u, v) = 0$ need not be symmetric in $u$ and $v$; it is symmetric for symmetric and alternating forms. In a basis that is orthogonal for $B$, the matrix of $B$ is diagonal. Unlike in an inner product space, a nonzero vector can be orthogonal to itself: for $B(x, y) = x_1 y_1 - x_2 y_2$, take $v = (1, 1)$.

Explanations

L1 human

Orthogonality makes sense for every bilinear form, not only for inner products. Two vectors are orthogonal when the form gives zero. A basis of pairwise orthogonal vectors makes the matrix of the form diagonal.