Linear Algebra
Differential Equations Math 54 Lec 005 (Dis 501) July 22, 2014
1 Quadratic Forms
1.1 Change of Variable in a Quadratic Form
Given any basis B={v1,· · · , vn}of Rn, let P=
v1v2· · · vn
. Then,
y=P−1x
is the B-coordinate of x. So, P, kind of, changes a variable into another variable.
Now, let Abe a symmetric matrix and define a quadratic form xTAx. Take Pas the matrix of which columns are
eigenvectors. (This can be done by the Spectral Theorem.) Especially, Phas orthonormal column vectors. Hence,
xTAx=yT(PTAP )y
and PTAP is a diagonal matrix of which diagonal entries are the eigenvalues of A. Such quadratic form can be written as
λ1y2
1+λ2y2
2+· · · +λny2
n.
1.2 Classifying Quadratic Forms
1.2.1 Positive Definite
Q(x)>0 for all x6=0
Furthermore, if Q(x)≥0 for all x6=0, then Qis positive semidefinite.
1.2.2 Negative Definite
Q(x)<0 for all x6=0
Furthermore, if Q(x)≤0 for all x6=0, then Qis negative semidefinite.
1.2.3 Indefinite
Q(x) has both positive and negative values.
In fact, the quadratic form is
positive definite if and only if the eigenvalues of Aare all positive
negative definite if and only if the eigenvalues of Aare all negative
indefinite if and only if Ahas both positive and negative eigenvalues.
Furthermore, it is positive semidefinite if and only if the eigenvalues of Aare nonnegative, negative semidefinite if and
only if the eigenvalues of Aare nonpositive.
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