# Theorems

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Professor 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 vAv =λvvand also
vAv =vAv=(Av)v=(λv)Tv=¯
λvv.Thus λ=¯
λ,i.e. λis real.
Theorem (Main theorem of symmetric matrices).Areal matrix Ais sym-
metric if and only if A=P1DPfor an orthogonal matrix Pand diagonal
matrix D,i.e. if and only if thereexists an orthonormal basis of eigenvectors
of A.
Proof. If A=P1DPfor an orthogonal matrix Pand diagonal matrix D,
then AT=PTDTP1T.Since PT=P1and Dis symmetric (since it is
diagonal), AT=P1DP=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.
1
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