Square

Several entities is described assquare in reference to the mathematician Charles Hermite.

Synopsis 1 Square operator 2 Square scalar product 3 Square matrix 4 Orthogonal polynomials of Hermit

Square operator

An operator is known as square if in a orthonormée base it equal to is transposed of combined sound (car-assistant).

That is to say:

With⁺ = ^T(A)^*

therefore, if

With = A⁺,

A is square.

Square scalar product

See scalar product.

It is said that a form is sesquilinéaire if (noting X, Y, Z of the vectors, and has, B of the scalars, i.e. complex numbers):

• <maths> \ f(aX+Y, Z)=\overline{a}f(X, Z)+f(Y, Z)</maths>, and
• <maths> \ f(X, bY+Z)=bf(X, Y)+f(X, Z)</maths>.
• Such a form is known as square if moreover <maths>f(X, Y)=\overline{f(Y, X)}</maths>.

It is it which intervenes for example in the spectral decomposition of Fourier.

A scalar product is square.

Square matrix

One square (or assistant car) is a square matrix with complex elements which checks the following property:

• the matrix is equal to the combined transposed matrix.

In other words

<maths>a_{i, J} = \overline{a_{j, i}}</maths>

For example,

<maths>\begin{bmatrix}3&2+i \ \

2-i&1\end{bmatrix}</maths>

is a square matrix.

In particular, a matrix with real elements is square if and only if it is .

A square matrix is diagonalisable and all its eigenvalues are real; its own subspaces are 2 to 2 orthogonal.

The square operators play an important part in quantum mechanics. They represent the physical sizes. The eigenvalues (real) represent the possible values of the size and the clean functions (or vectors) the associated states.

Orthogonal polynomials of Hermit

The continuation of polynomials of Hermit, noted <maths>H_n</maths>, is orthogonal for the scalar product <maths> <f, g> = \int_{-\infty}^{+\infty} {f(x)g(x)e^{-x^2}}{dx} </maths>.

These polynomials are defined in such a way that <maths>H_n</maths> maybe of degree N, the initial term being <maths>H_0 = 1</maths>.

This continuation satisfies the following relations:

• <maths>H^{}_{n+1}(x)-2xH_n(x)+2nH_{n-1}(x)=0</maths>
• <maths>H^{'}_n(x)=2nH_{n-1}(x)</maths>
• <maths>H_ñ{} - 2x.H_ñ{'} + 2nH_n = 0</maths>
• <maths>H_n(x)=(-1)^{n}e^{x^2}\frac{d^n}{dx^n}(e^{-x^2})</maths>

The polynomials of Hermit intervene in the theory of the uniform approximation of the functions. In physics, one finds them in the resolution ofequation of heat, but also in where they give them functions of waves harmonic oscillator.