Definition:Orthogonal (Linear Algebra)
Definition
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $u, v \in V$.
We say that $u$ and $v$ are orthogonal if and only if:
- $\innerprod u v = 0$
We denote this:
- $u \perp v$
Orthogonal Set
Let $S = \set {u_1, \ldots, u_n}$ be a subset of $V$.
Then $S$ is an orthogonal set if and only if its elements are pairwise orthogonal:
- $\forall i \ne j: \innerprod {u_i} {u_j} = 0$
Orthogonality of Sets
Let $A, B \subseteq V$.
We say that $A$ and $B$ are orthogonal if and only if:
- $\forall a \in A, b \in B: a \perp b$
That is, if $a$ and $b$ are orthogonal elements of $A$ and $B$ for all $a \in A$ and $b \in B$.
We write:
- $A \perp B$
Orthogonal Complement
Let $S\subseteq V$ be a subset.
We define the orthogonal complement of $S$ (with respect to $\innerprod \cdot \cdot$), written $S^\perp$ as the set of all $v \in V$ which are orthogonal to all $s \in S$.
That is:
- $S^\perp = \set {v \in V : \innerprod v s = 0 \text { for all } s \in S}$
If $S = \set v$ is a singleton, we may write $S^\perp$ as $v^\perp$.
Vectors in $\R^n$
Let $\mathbf u$, $\mathbf v$ be vectors in $\R^n$.
Then $\mathbf u$ and $\mathbf v$ are said to be orthogonal if and only if their dot product is zero:
- $\mathbf u \cdot \mathbf v = 0$
As Dot Product is Inner Product, this is a special case of the definition of orthogonal vectors.
Also see
- Results about orthogonality in the context of Linear Algebra can be found here.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Definition $2.1$
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $9.2$: Orthonormal Sets