Linear and multilinear algebra; matrix theory Definition

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 $v = (1, 1)$ .

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Vector space definition
Bilinear form definition
Orthogonal family for a bilinear form definition

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Tag	`05Q4JD6XEH`
URI	`m2://05Q4JD6XEH`
Kind	definition
Writing	W0 stub
MSC	15A63
Defines	Orthogonal family for a bilinear form
Uses	Bilinear form

Orthogonal family for a bilinear form

Definition

Remarks

Reading path

Where this leads

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