Polarization identity

In linear algebra, a symmetric bilinear or a sesquilinear form is represented using its associated quadratic form by a polarization formula.

An important application is the representation of the scalar product of an inner product space by the corresponding induced norm. Conversely, one can ask whether a given norm is induced by an inner product. This is exactly the case if the norm satisfies the parallelogram law, the dot product can then be determined by means of polarization of the square of the norm.

The real case ( symmetric form )

Let a vector space over the field and a symmetric bilinear form, ie

For everyone.

Its corresponding square shape is then defined by

Conversely, the symmetric bilinear form is uniquely determined by its square shape. This pushes the polarization formula of: it is

That this does not apply to any (also non-symmetric ) bilinear forms, the following example shows. With the aid of matrices

Are the bilinear forms given by

Then, and different, but define the same square shape.

The complex case ( sesquilinear )

Let a vector space over the field and a sesquilinear. Its associated quadratic form is defined as in the real case by

Also a sesquilinear form is uniquely determined by its square shape. For Sesquilinearformen is the polarization formula:

If the first argument is semilinear and

Is semilinear if the second argument.