ThmDex – An index of mathematical definitions, results, and conjectures.

Processing math: 58%

Result R249 on D63: Finite sequence

Cauchy-Schwarz inequality for finite real sequences

Formulation 0

Let

$x_1, y_1, \ldots, x_N, y_N \in \mathbb{R}$ each be a D993: Real number.

Then

(1)	$\begin{equation} \left\| \sum_{n = 1}^N x_n y_n \right\| \leq \left( \sum_{n = 1}^N x_n \right)^{1/2} \left( \sum_{n = 1}^N y_n \right)^{1/2} \end{equation}$
(2)	$\begin{equation} \left\| \sum_{n = 1}^N x_n y_n \right\| = \left( \sum_{n = 1}^N x_n \right)^{1/2} \left( \sum_{n = 1}^N y_n \right)^{1/2} \quad \iff \quad \frac{x_1}{y_1} = \frac{x_2}{y_2} = \cdots = \frac{x_N}{y_N} \end{equation}$

Formulation 1

Let

$x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then

(1)	$\begin{equation} \left\| x^T y \right\| \leq \sqrt{x^T x} \sqrt{y^T y} \end{equation}$
(2)	$\begin{equation} \left\| x^T y \right\| = \sqrt{x^T x} \sqrt{y^T y} \quad \iff \quad \frac{x_1}{y_1} = \frac{x_2}{y_2} = \cdots = \frac{x_N}{y_N} \end{equation}$

Formulation 2

Let

$x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then

(1)	$\begin{equation} \left\| x^T y \right\| \leq \Vert x \Vert_2 \Vert y \Vert_2 \end{equation}$
(2)	$\begin{equation} \left\| x^T y \right\| = \Vert x \Vert_2 \Vert y \Vert_2 \quad \iff \quad \frac{x_1}{y_1} = \frac{x_2}{y_2} = \cdots = \frac{x_N}{y_N} \end{equation}$

Formulation 3

Let

$x, y \in \mathbb{R}^{N \times 1}$ each be a D5200: Real column matrix.

Then

(1)	$\begin{equation} \left\| x \cdot y \right\| \leq \sqrt{x \cdot x} \sqrt{y \cdot y} \end{equation}$
(2)	$\begin{equation} \left\| x \cdot y \right\| = \sqrt{x \cdot x} \sqrt{y \cdot y} \quad \iff \quad \frac{x_1}{y_1} = \frac{x_2}{y_2} = \cdots = \frac{x_N}{y_N} \end{equation}$

Also known as

Cauchy-Schwarz inequality for real column matrices