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| Orthogonality generalises perpendicularity to an [[inner product space]].
| | The Kronecker delta function is a mathematical operation used to select or compare indices. |
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| == Mathematical definition == | | == Mathematical definition == |
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| Let $V$ be an [[inner product space]].
| | === Classical version === |
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| If for $u,v\in V$
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| $$\langle u, v \rangle=0$$
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| Then $u,v$ are '''orthogonal''' to each other.
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| $u\bot v$ denotes this orthogonality.
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| == Application == | |
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| === Orthogonal decomposition === | |
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| Given vectors $u,v\in V$ and $v\neq 0$, $u$ can be uniquely decomposed into two pieces: one parallel to $v$ and one orthogonal to $v$. This is called an '''orthogonal decomposition'''.
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| In mathmetical terms:
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| $$u=u_1+u_2,$$
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| Where $u_1=a v, a \in \mathbb{F}; u_2\bot v$
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| '''Derivation:'''
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| $$ | | $$ |
| \because u = u_1 +u_2 \\ | | \delta_{ij} = |
| \therefore u_2=u-u_1=u-av \\
| | \begin{cases} |
| \because u_2 \bot v \\
| | 1, & i=j,\\ |
| \therefore \langle u_2, v \rangle = \langle u- av, v \rangle = \langle u, v \rangle - a \left\| v \right\|^2 = \vec 0 \\
| | 0, & i\ne j. |
| \text{Solve for }a,a = \frac{\langle u, v \rangle}{\left\| v \right\|^2} \\
| | \end{cases} |
| \text{Substitute back into }u_1=av, u_2=u-av, \\
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| u=\frac{\langle u, v \rangle}{\| v \|^2} v + \left( u-\frac{\langle u, v \rangle}{\| v \|^2} v \right)
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| $$ | | $$ |
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| == Properties == | | === single-index variant === |
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| === $\vec 0$ is orthogonal to every vector ===
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| $$ | | $$ |
| \forall v \in V, \vec 0 \bot v | | \delta_n = |
| | \begin{cases} |
| | 1, & n=0,\\ |
| | 0, & n\ne 0. |
| | \end{cases} |
| $$ | | $$ |
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| '''Proof''':
| | == Properties == |
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| Recall [[Vector_space#Mathematical_definition|the properties of a vector space by definition]]. By additive identity,
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| | === Symmetry === |
| $$ | | $$ |
| \vec 0 +\vec 0 = \vec 0 | | \delta_{ij} = \delta_{ji}. |
| $$ | | $$ |
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| Therefore,
| | == Application == |
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| $$
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| \langle \vec 0, u \rangle = \langle \vec 0+\vec 0, u\rangle
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| $$
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| Recall the [[inner product|properties of the inner product by definition]], by linearity in the 1st slot,
| | === Single out the $i$-th term of a sum === |
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| $$ | | $$ |
| \langle \vec 0+\vec 0, u\rangle = \langle\vec 0, u \rangle+\langle \vec 0, u\rangle | | \sum_j \delta_{ij} a_j = a_i. |
| $$
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| Subtracting $\langle \vec 0, u\rangle$ from both sides gives
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| $$ | | $$ |
| \langle \vec 0, u \rangle=0.
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| $$
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| === Generalised Pythagorean theorem ===
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| ''If'' $u,v\in V; u\bot v$, then $\| \cdot \|$ defined by $\| v \|:=\sqrt{\langle v, v \rangle}$ obeys
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| $$\| u+v \|^2 = \| u \|^2 + \| v \|^2.$$
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| '''Proof:''':
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| By definition,
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| $$
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| \| u+v \|^2 = \langle u+v, u+v \rangle
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| $$
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| Recall the [[inner product|properties of the inner product by definition]]:
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| By linearity in the 1st slot,
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| $$
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| \langle u+v, u+v \rangle = \langle u, u+v \rangle + \langle v, u+v \rangle
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| $$
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| By conjugate symmetry,
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| $$
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| \langle u, u+v \rangle + \langle u, u+v \rangle = \overline{\langle u+v, u \rangle} + \overline{\langle u+v, v \rangle}
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| $$
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| Again by linearity in the first slot,
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| $$
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| \overline{\langle u+v, u \rangle} + \overline{\langle u+v, v \rangle}=\overline{\langle u, u \rangle} + \overline{\langle v, u \rangle} + \overline{\langle u, v \rangle} + \overline{\langle v, v \rangle}
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| $$
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| Again by conjugate symmetry,
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| $$
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| \overline{\langle u, u \rangle} + \overline{\langle v, u \rangle} + \overline{\langle u, v \rangle} + \overline{\langle v, v \rangle} = \langle u, u \rangle + \langle u, v \rangle + \langle v, u \rangle + \langle v, v \rangle
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| $$
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| since $u,v\in V; u\bot v$:
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| $$
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| \langle u, v \rangle=0;\langle v, u \rangle=0
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| $$
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| By definition
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| $$
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| \langle u, u \rangle=\left\| u \right\|^2;\langle v, v \rangle=\left\| v \right\|^2
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| $$
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| Hence
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| $$\| u+v \|^2 = \| u \|^2 + \| v \|^2.$$
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| == Orthogonal vector list ==
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| A list of '''nonzero''' vectors $(e_1,\ldots,e_m) \in V$ is '''orthogonal''' if
| | === Identity matrix === |
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| $$ | | $$ |
| \langle e_i, e_j \rangle = 0 \Leftarrow 1\le i\neq j \le m
| | I_{ij} = \delta_{ij}. |
| $$ | | $$ |
The Kronecker delta function is a mathematical operation used to select or compare indices.
Mathematical definition
Classical version
$$
\delta_{ij} =
\begin{cases}
1, & i=j,\\
0, & i\ne j.
\end{cases}
$$
single-index variant
$$
\delta_n =
\begin{cases}
1, & n=0,\\
0, & n\ne 0.
\end{cases}
$$
Properties
Symmetry
$$
\delta_{ij} = \delta_{ji}.
$$
Application
Single out the $i$-th term of a sum
$$
\sum_j \delta_{ij} a_j = a_i.
$$
Identity matrix
$$
I_{ij} = \delta_{ij}.
$$