Self Adjoint Linear Maps

If V is a vector space of 2 dimensions and an inner product given by < , > we sat that a linear map A : V --> V is self-adjoint if <Av,w>=<v,Aw> for all v and w in V. Note that if {e₁,e₂} is an orthonormal basis for V and (a_ij), i,j=1,2, is the matrix of A relative to that basis, then

<Ae_i,e_j> = a_ij = <e_i,Ae_j> = <Ae_j,e_i> = a_ji
and (a_ij) is symmetric.

To each self-adjoint map A we associate a map B : V x V --> R defined by B(v,w) = <Av,w>. Because of the definition of the inner product and the properties of matrixes, B is linear in both v and w. Further, since A is self-adjoint, B(v,w) = B(w,v). Therefore, B is a bilinear symmetric form in V.
On the other hand, if B is a bilinear symmetric form in V, we can define a linear map A : V --> V by <Av,w> = B(v,w) and the symmetry of B implies that A is self-adjoint.

 

Quadratic Forms

For each symmetric bilinear form B in V, there corresponds a quadratic form Q in V given by Q(v) = B(v,v) for v in V. This relationship is one-to-one, since

B(u,v) =

[Q(u+v)-Q(u)-Q(v)] 2

Lemma: If the function Q(x,y) = ax²+2bxy+cy², restricted to the unit circle, has a maximum at the point (1,0), then b = 0. (proof)

Now, using the lemma, we can construct the following proposition:
Given a quadratic form Q in V, there exists an orthonormal basis {e₁,e₂} of V such that if v in V is given by v = x e₁ + y e₂, then

Q(v) = l₁ x² + l₂ y²,

where l₁ and l₂ are the maximum and minimum, respectively, of Q on the unit circle |v| = 1. (proof)

We say that a vector v != 0 is an eignevector of a linear map A : V --> V if Av = lv for some real number l; l is then called an eigenvalue of A.

The previous definitions allow us to prove the following theorem:
Let A : V --> V be a self-adjoint linear map. Then there exists an orthonormal basis {e₁,e₂} of V such that A(e₁) = l₁e₁, A(e₂) = l₂e₂ (that is, e₁ and e₂ are eigenvectors, and l₁ and l₂ are eigenvalues of A). In the basis {e₁,e₂}, the matrix of A is clearly diagonal adn the elements l₁, l₂, l₁ > l₂, on the diagonal are the maximum and the minimum, respectively, of the quadratic form Q(v) = <Av,v> on the unit circle of V. (proof)