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IV. A Natural Physical Interpretation

It is now important to give a hint of physical interpretation for the external factor k occurring when one wants to construct the cubic given its Hessian. For this we will consider the case where the cubic has real roots, i.e. k is complex of unit modulus. The equations (3) for the roots can be written as

with h, h^* the roots of Hessian and w º (-1 + i×Ö3) /2 the cubic root of unity [i º Ö(-1)]. Now consider the vector of components x₁, x₂, x₃. This is a vector with respect to a special group (See the comment following (Baker, 1901) in REFERENCES) to be mentioned later, but it just happens to represent a real Space situation when the three roots are the principal values of a symmetric matrix. We are legitimate in using this image, for there is a Space reference frame we can construct in every point of Space where the symmetric matrix is defined, as given by three special orthogonal vectors – the principal directions of the said symmetric matrix. Thus the principal values of such a matrix can be represented by the column matrix

which is plainly a vector in matrix representation, for each principal value can be interpreted as the component of the vector along the corresponding principal direction. We can decompose this vector with respect to the plane cutting the axes of reference frame in the points situated at unit distance from origin. In engineering applications this plane is called octahedral plane, for it represents the face of an octahedron in space. The normal component of the vector (36) on this plane is, with an obvious notation for transposed vectors, given by

The in-plane (tangential) component of (36) is then given by

It is this last vector, usually called octahedral shear vector in engineering applications, which allows us to interpret the complex number k externally introduced. Namely, the Sylvester form (2) of our cubic allows us to identify its binomial coefficients in terms of the quantities h, h^* and k, up to an arbitrary factor, as

From this we have right away

Now take as reference the vector corresponding to k = 1, when the roots are solely determined by the roots of Hessian

and then calculate the angle of the generic tangential vector with respect to this one by the well known formula:

Using equations (40) and (41) we have

so that equation (42) becomes

We conclude that, indeed, knowing the Hessian does not determine the cubic uniquely. In the case of a cubic with real roots the Hessian actually determines a family of cubics whose roots are defined as a one parameter family. The parameter of this family is given by the angle of orientation of the octahedral shear vector in the octahedral plane.

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