262 Orthonormal
A property of a set of vectors of unit norm whose pairwise inner products vanish when the vectors differ that is used to build a convenient basis.
definition [D] (Orthonormal) A collection of mutually orthogonal normalized vectors in an inner product space. The set must satisfy the following conditions:
- Each vector in the set has unit length.
- Any two distinct vectors in the set are orthogonal, meaning their inner product is zero.
where
- \(\langle \alpha_i | \alpha_j \rangle\) is the inner product of two vectors in the set.
- \(\delta_{ij}\) is the Kronecker delta, which is 1 if \(i = j\) and 0 otherwise.
- \(\|\alpha\|\) is the norm of a vector, defined as the square root of the inner product of the vector with itself.
- \(V\) is an inner product space.
Note:
- unit length means \(\|\alpha\| = 1\).
- the norm is also called the length.
- \(V\) may be regarded as a vector space with an inner product.
262.1 Elementary Example
262.1.1 Simple
An orthonormal set has unit vectors that are pairwise orthogonal.
\[ B = \{ e_{1},\ e_{2} \} \]
\[ e_{1} = (1,0),\quad e_{2} = (0,1) \]
\[ \langle e_{i}, e_{j} \rangle = \delta_{ij} \]
where
- \(\langle e_{i}, e_{i} \rangle = 1\).
- \(\langle e_{1}, e_{2} \rangle = 0\).
- \(\delta_{ij}\) is the Kronecker delta.
262.1.2 General
The standard basis of \(\mathbb{R}^{3}\) is orthonormal.
\[ B = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \langle e_{i}, e_{j} \rangle = \delta_{ij},\quad i,j \in \{1,2,3\} \]
where
- each \(\lVert e_{i} \rVert = 1\).
- Adjoint
- Basis
- Characteristic Polynomial
- Complex Conjugate
- Conjugate Symmetry
- Conjugate Transpose
- Determinant
- Diag
- Dual Space
- Eigendecomposition
- Eigenvalue
- Eigenvector
- Hermitian
- Hermitian Conjugate
- Homogeneity
- Homogeneous
- Inner Product
- Inner Product Space
- Kronecker Delta
- Linear Function
- Linear Map
- Linear Transformation
- Operator
- Orthonormal
- Orthonormal Set of Functions
- Projection Map
- Quadratic
- Standard Basis