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Lecture 11 notes byY. Burda

Wediscussed the following theorems:

Theorem. Let Abe areal symmetric matrix. Then allits eigenvalues are

real.

Proof. If Av =λv for some complexλand v,then v∗Av =λv∗vand also

v∗Av =v∗A∗v=(Av)∗v=(λv)Tv=¯

λv∗v.Thus λ=¯

λ,i.e. λis real.

Theorem (Main theorem of symmetric matrices).Areal matrix Ais sym-

metric if and only if A=P−1DPfor an orthogonal matrix Pand diagonal

matrix D,i.e. if and only if thereexists an orthonormal basis of eigenvectors

of A.

Proof. If A=P−1DPfor an orthogonal matrix Pand diagonal matrix D,

then AT=PTDTP−1T.Since PT=P−1and Dis symmetric (since it is

diagonal), AT=P−1DP=A,i.e. Ais symmetric.

For the other direction wefollowed the book (theorem 8.19 on p.325)

prettyclosely.

The main corollary of this theorem for us will bethe following theorem:

Theorem (Principal axis theorem).Let qbe areal quadratic form on Rn.

Then thereexists an orthonormal basis of Rnso that in coordinates y1,...,yn

relative to this basis the form qcan be written as λ1y2

1+. . . +λny2

n.

It is called “principal axis theorem” because it establishes the existence

of nmutually orthogonal axesfor anyquadric surface, anologues ofminor

and major axes of an ellipse.

One importantapplication of principal axis theorem is to the inertia ten-

sor of arigid body.It turns out that to anythree-dimensional body one can

associate aquadratic form called inertia tensor. Its principal axes havethe

propertythat the body can revolvearound these axes (passed through the

center of mass) without anyrotation ofthe axes themselves. For all other

modes ofrotation, the axis of rotation is itself moving.

Even though theprincipal axis theorem is just an existence theorem —

it doesn’t simplify calculations in concrete problems —it can still bevery

useful:

Example

Determine what is the shapedescribed bythe equation x2+8xy +7y2=1.

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