Orthogonality | Sahithyan's Notes

Ortho is a Greek prefix meaning right.

Definition

When their inner product is $0$ . Angle between the vectors are $90^\circ$ .

When all pairs of vectors are orthogonal.

An orthogonal set is automatically linearly-independent.

When 2 vectors or all pairs of vectors of a set are:

A method to convert a linearly independent set into a orthogonal set.

Suppose $\set{x_1,\dots,x_n}$ is linearly independent.

y_k = x_k - \sum_{j<k} \frac{\langle x_k, y_j\rangle}{|y_j|^2} y_j.

For $\langle p,q\rangle = \int_{-1}^{1} p(x)q(x)\,\text{d}x$ , its orthogonal set is named Legendre polynomials of the first kind.

\set{ p_n(x) } = \bigg\{ 1, x, \frac{1}{2}(3x^2-1), \dots \bigg\}

$n$ starts from 0. Additionally $p_n(x)$ also satisfies:

(1-x^2)y'' - 2xy' + x(x+1)y = 0

For $\langle p,q\rangle = \int_{0}^{\infty} e^{-x}p(x)q(x)\,\text{d}x$ , its orthogonal set is named Leguerre polynomials of the first kind.

\set{ L_n(x) } = \set { 1, 1-x, \frac{1}{2}(x^2-4x+2) }

$n$ starts from 0. Additionally $L_n(x)$ satisfies:

xy'' - (1-x)y' + xy = 0