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Dot product or Inner product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.


Algebraic definition

Let \(X\) and \(Y\) be vectors of \(\mathbb{R}^n\): \(X=(X_1, X_2, ..., X_n)\) and \(Y=(Y_1, Y_2, ..., Y_n)\). In an Euclidian space, the dot product is the sum of the products of the corresponding cartesian coordinates of \(X\) and \(Y\).

\[\langle X, Y \rangle = \sum_{i=1}^n X_i \cdot Y_i\]

\(\langle X, Y \rangle\) can also be written \(\langle X \vert Y \rangle\) or \(X \cdot Y\).


Geometric definition

Dot product of \(X\) and \(Y\) can also be defined using the angle \(\theta\) between the two vectors and their lenghts (obtain using theirs euclidian norms):

\[\langle X, Y \rangle = \Vert X \Vert \Vert Y \Vert \cos \theta\]


Property

If the angle between \(X\) and \(Y\) is 90° (or equivalently \(\frac{\pi}{2}\)) then \(X\) and \(Y\) are orthogonal and:

\[\langle X, Y \rangle = \Vert X \Vert \Vert Y \Vert \cos \left(\frac{\pi}{2}\right) = 0\]


Norm

The notion of norm is the extension of absolute value in dimension greater than 1.

It defines a length for vectors and can be used as a distance measure.

Different norm exists, the natural norm being the euclidian norm.


Types of norm

Let \(X\) be a vector of \(\mathbb{R}^n\): \(X=(X_1, X_2, ..., X_n)\).


Manhattan norm (norm 1)

\[\Vert X \Vert_1 = \sum_{i=1}^n \vert X_i \vert\]

The norm 1 of a vector is the sum of the absolute value of each of its composants.


Euclidian norm (norm 2)

\[\Vert X \Vert_2 = \sqrt{\sum_{i=1}^n X_i^2}\]

The Euclian norm of a vector is the square root of the sum of the composants of \(X\) squared.


The Euclidian norm can also be defined as the square root of the dot product of itself and itself:

\[\Vert X \Vert_2 = \sqrt{\langle X, X \rangle}\]

Or:

\[\Vert X \Vert_2^2 = \langle X, X \rangle\]


\(p\)-norm

\[\Vert X \Vert_p = \left( \sum_{i=1}^n X_i^p \right)^{1/p}\]


Maximum norm (or infinity norm of supremum norm)

\[\Vert X \Vert_\infty = \max(\vert X_1 \vert, \vert X_2 \vert, ..., \vert X_n \vert)\]

The maximum norm of a vector is the maximum of the absolute value of its composants.


Resources

See:


Determinant


Characteristic polynomial

A polynomial can be factorized using its roots: if \(\lambda_i\) are root of \(f\) then:

\[f(x)=\prod_{i=1}^{n_{roots}}(x-\lambda_i)\]


Resources

See:


Diagonalisation