# Square

Several entities Mathématiques are described as square in reference to the mathematician Charles Hermite.

## Square space

One calls square space any space vector E complex of finished size provided with a scalar Produit square.

## Square scalar product

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

• $\ F \left(aX+Y, Z\right) = \ overline \left\{has\right\} F \left(X, Z\right) +f \left(Y, Z\right)$, and
• $\ F \left(X, bY+Z\right) =bf \left(X, Y\right) +f \left(X, Z\right)$.
• Such a form is known as square so moreover $f \left(X, Y\right) = \ overline \left\{F \left(Y, X\right)\right\}$.
• It is known as square definite positive if $f \left(X, X\right) >0 \,$ for any vector $X \, \ not=0 \,$.

A square scalar product is a positive definite square form.

The two basic examples are $\ mathbb \left\{C\right\} ^n \,$, with

$F \left(U, V\right) = \ sum_ \left\{i=1\right\} ^n \ overline \left\{u_i\right\} \left\{v_i\right\}$
and $L^2 \left(I\right) \,$ for an interval $I \ subset \ R \,$, with
$F \left(G, H\right) = \ int_I \ overline \left\{G \left(T\right)\right\}H \left(T\right) dt$
(One considers functions with complex values.)

In theory of the Fourier series, it is more convenient to work with the $e^ \left\{inx\right\} \,$ that with the sines and the cosine, which explains the intervention of this concept in the spectral decomposition of Fourier.

The two basic properties of the real scalar product remain:

• the Inequality of Cauchy-Schwarz;
• $\ sqrt \left\{F \left(X, X\right)\right\}$ is a standard (a consequence).

## Square operator

An operator U of square space E is known as square if:

$\ forall X \ in E, \ forall there \ in E, \left(U \left(X\right)|there\right) = \left(X|U \left(there\right)\right)$

The square operators play a big role in quantum Mécanique, because they represent the physical sizes. The eigenvalues (real) represent the possible values of the size and the clean functions (or vectors) the associated states.

In a orthonormal Base, the matrix of such an operator is equal to the Transposée of sound Conjugué (car-assistant). Let us note: $A^+ = \left\{\right\} ^t \left(A\right)^*$. Then if $A = A^+$, has is the matrix of a square operator.

## Square matrix

A square matrix (or car-assistant) is a square matrix with complex elements which checks the following property:
• the matrix is equal to the combined transposed matrix.
In other words, $a_ \left\{I, J\right\} = \ overline \left\{a_ \left\{J, I\right\}\right\}$

For example, $A= \ begin \left\{pmatrix\right\} 3&i&-5i \ \ - i&-2&5 \ \ 5i&5&10\end\left\{pmatrix\right\}$ is a square matrix.

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

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

## orthogonal Polynomials of Hermit

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

The continuation of the Polynomials of Hermit, noted $H_n$, is orthogonal for the scalar Produit defined by:

$< \! F, G \! > \, = \ int_ \left\{- \ infty\right\} ^ \left\{+ \ infty\right\} \left\{F \left(X\right) \, G \left(X\right) \, \ mathrm \left\{E\right\} ^ \left\{- x^2\right\}\right\} \left\{dx\right\}$.

These polynomials are defined in such a way that $H_n$ is of degree N, the first of them being $H_0 =1$.

This continuation satisfies the following relations:

• $H^ \left\{\right\} _ \left\{n+1\right\} \left(X\right) - 2x \, H_n \left(X\right) +2n \, H_ \left\{n-1\right\} \left(X\right) =0$
• $H^ \left\{“\right\} _n \left(X\right) =2n \, H_ \left\{n-1\right\} \left(X\right)$
• $H_n^ \left\{$} - 2x \, H_n^ {”} + 2n \, H_n = 0
• $H_n \left(X\right) = \left(- 1\right) ^ \left\{N\right\} \, \ mathrm \left\{E\right\} ^ \left\{x^2\right\} \ frac \left\{d^n\right\} \left\{dx^n\right\} \ left \left(\ mathrm \left\{E\right\} ^ \left\{- x^2\right\}\ right\right)$

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