# Sylvester's Law of Inertia and Applications

DEFINITION 6.4.1 (Bilinear Form)   Let be a matrix with real entries. A bilinear form in is an expression of the type

Observe that if (the identity matrix) then the bilinear form reduces to the standard real inner product. Also, if we want it to be symmetric in and then it is necessary and sufficient that for all Why? Hence, any symmetric bilinear form is naturally associated with a real symmetric matrix.

DEFINITION 6.4.2 (Sesquilinear Form)   Let be a matrix with complex entries. A sesquilinear form in is given by

Note that if (the identity matrix) then the sesquilinear form reduces to the standard complex inner product. Also, it can be easily seen that this form is linear' in the first component and conjugate linear' in the second component. Also, if we want then the matrix need to be an Hermitian matrix. Note that if and , then the sesquilinear form reduces to a bilinear form.

The expression is called the quadratic form and the Hermitian form. We generally write and in place of and , respectively. It can be easily shown that for any choice of the Hermitian form is a real number.

Therefore, in matrix notation, for a Hermitian matrix , the Hermitian form can be rewritten as

EXAMPLE 6.4.3   Let Then check that is an Hermitian matrix and for the Hermitian form

where Re' denotes the real part of a complex number. This shows that for every choice of the Hermitian form is always real. Why?

The main idea is to express as sum of squares and hence determine the possible values that it can take. Note that if we replace by where is any complex number, then simply gets multiplied by and hence one needs to study only those for which i.e., is a normalised vector.

From Exercise 6.3.11.3 one knows that if ( is Hermitian) then there exists a unitary matrix such that ( with 's the eigenvalues of the matrix which we know are real). So, taking (i.e., choosing 's as linear combination of 's with coefficients coming from the entries of the matrix ), one gets

 (6.4.1)

Thus, one knows the possible values that can take depending on the eigenvalues of the matrix in case is a Hermitian matrix. Also, for represents the principal axes of the conic that they represent in the n-dimensional space.

Equation (6.4.1) gives one method of writing as a sum of absolute squares of linearly independent linear forms. One can easily show that there are more than one way of writing as sum of squares. The question arises, what can we say about the coefficients when has been written as sum of absolute squares".

This question is answered by Sylvester's law of inertia' which we state as the next lemma.

LEMMA 6.4.4   Every Hermitian form (with an Hermitian matrix) in variables can be written as

where are linearly independent linear forms in and the integers and depend only on

Proof. From Equation (6.4.1) it is easily seen that has the required form. Need to show that and are uniquely given by

Hence, let us assume on the contrary that there exist positive integers with such that /

Since, and are linear combinations of we can find a matrix such that Choose . Since Theorem 2.5.1, gives the existence of finding nonzero values of such that Hence, we get

Now, this can hold only if which gives a contradiction. Hence

Similarly, the case can be resolved. height6pt width 6pt depth 0pt

Note: The integer is the rank of the matrix and the number is sometimes called the inertial degree of

We complete this chapter by understanding the graph of

for We first look at the following example.

EXAMPLE 6.4.5   Sketch the graph of

Solution: Note that

The eigenpairs for are Thus,

Let

Then

Thus the given graph reduces to

Therefore, the given graph represents an ellipse with the principal axes and That is, the principal axes are

The eccentricity of the ellipse is the foci are at the points and and the equations of the directrices are

DEFINITION 6.4.6 (Associated Quadratic Form)   Let be the equation of a general conic. The quadratic expression

is called the quadratic form associated with the given conic.

We now consider the general conic. We obtain conditions on the eigenvalues of the associated quadratic form to characterise the different conic sections in (endowed with the standard inner product).

PROPOSITION 6.4.7   Consider the general conic

Prove that this conic represents
1. an ellipse if
2. a parabola if and
3. a hyperbola if

Proof. Let Then the associated quadratic form

As is a symmetric matrix, by Corollary 6.3.7, the eigenvalues of are both real, the corresponding eigenvectors are orthonormal and is unitarily diagonalisable with

 (6.4.2)

Let Then

and the equation of the conic section in the -plane, reduces to

Now, depending on the eigenvalues we consider different cases:

1. Substituting in (6.4.2) gives Thus, the given conic reduces to a straight line in the -plane.

2. In this case, the equation of the conic reduces to

1. If then in the -plane, we get the pair of coincident lines .
2. If
1. If then we get a pair of parallel lines
2. If the solution set corresponding to the given conic is an empty set.
3. If Then the given equation is of the form for some translates and and thus represents a parabola.

Also, observe that implies that the That is,

3. and
Let Then the equation of the conic can be rewritten as

In this case, we have the following:
1. suppose Then the equation of the conic reduces to

The terms on the left can be written as product of two factors as Thus, in this case, the given equation represents a pair of intersecting straight lines in the -plane.
2. suppose As we can assume So, the equation of the conic reduces to

This equation represents a hyperbola in the -plane, with principal axes

As we have

4. In this case, the equation of the conic can be rewritten as

we now consider the following cases:
1. suppose Then the equation of the ellipse reduces to a pair of perpendicular lines and in the -plane.
2. suppose Then there is no solution for the given equation. Hence, we do not get any real ellipse in the -plane.
3. suppose In this case, the equation of the conic reduces to

This equation represents an ellipse in the -plane, with principal axes

Also, the condition implies that

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Remark 6.4.8   Observe that the condition

implies that the principal axes of the conic are functions of the eigenvectors and

EXERCISE 6.4.9   Sketch the graph of the following surfaces:

As a last application, we consider the following problem that helps us in understanding the quadrics. Let

 (6.4.3)

be a general quadric. Then we need to follow the steps given below to write the above quadric in the standard form and thereby get the picture of the quadric. The steps are:
1. Observe that this equation can be rewritten as

where

2. As the matrix is symmetric matrix, find an orthogonal matrix such that is a diagonal matrix.
3. Replace the vector by Then writing the equation (6.4.3) reduces to

 (6.4.4)

where are the eigenvalues of
4. Complete the squares, if necessary, to write the equation (6.4.4) in terms of the variables so that this equation is in the standard form.
5. Use the condition to determine the centre and the planes of symmetry of the quadric in terms of the original system.

EXAMPLE 6.4.10   Determine the quadric
Solution: In this case, and and . Check that for the orthonormal matrix , So, the equation of the quadric reduces to

Or equivalently,

So, the equation of the quadric in standard form is

where the point is the centre. The calculation of the planes of symmetry is left as an exercise to the reader.

A K Lal 2007-09-12