Linear Algebra, Theory And Applications, 2012a

318 SELF ADJOINT OPERATORS
Corollary 13.4.2 Let X be a finite dimensional Hilbert space and let {v 1 , ··· ,v n } be an
orthonormal basis for X. Also, let q be the coordinate map associated with this basis satisfying
q (x) ≡ ∑ i x iv i . Then (x, y) F n =(q (x) ,q(y)) X
. Also, if A ∈L(X, X) , and M (A)
is the matrix of A with respect to this basis,
(Aq (x) ,q(y)) X
=(M (A) x, y) F n .
Definition 13.4.3 A self adjoint A ∈ L(X, X) , is positive definite if whenever x ≠ 0,
(Ax, x) > 0 and A is negative definite if for all x ≠ 0, (Ax, x) < 0. Ais positive semidefinite
or just nonnegative for short if for all x, (Ax, x) ≥ 0. Ais negative semidefinite or
nonpositive for short if for all x, (Ax, x) ≤ 0.
The following lemma is of fundamental importance in determining which linear transformations
are positive or negative definite.
Lemma 13.4.4 Let X be a finite dimensional Hilbert space. A self adjoint A ∈L(X, X)
is positive definite if and only if all its eigenvalues are positive and negative definite if and
only if all its eigenvalues are negative. It is positive semidefinite if all the eigenvalues are
nonnegative and it is negative semidefinite if all the eigenvalues are nonpositive.
Proof: Suppose first that A is positive definite and let λ be an eigenvalue. Then for x
an eigenvector corresponding to λ, λ (x, x) =(λx, x) =(Ax, x) > 0. Therefore, λ>0as
claimed.
Now suppose all the eigenvalues of A are positive. From Theorem 13.3.3 and Corollary
13.3.6, A = ∑ n
i=1 λ iu i ⊗ u i where the λ i are the positive eigenvalues and {u i } are an
orthonormal set of eigenvectors. Therefore, letting x ≠ 0,
(( n
) ) (
∑
n
)
∑
(Ax, x) = λ i u i ⊗ u i x, x = λ i u i (x, u i ) , x
=
i=1
(
∑ n
)
λ i (x, u i )(u i , x) =
i=1
i=1
n∑
λ i |(u i , x)| 2 > 0
because, since {u i } is an orthonormal basis, |x| 2 = ∑ n
i=1 |(u i, x)| 2 .
To establish the claim about negative definite, it suffices to note that A is negative
definite if and only if −A is positive definite and the eigenvalues of A are (−1) times the
eigenvalues of −A. The claims about positive semidefinite and negative semidefinite are
obtained similarly.
The next theorem is about a way to recognize whether a self adjoint A ∈L(X, X) is
positive or negative definite without having to find the eigenvalues. In order to state this
theorem, here is some notation.
Definition 13.4.5 Let A be an n × n matrix. Denote by A k the k × k matrix obtained by
deleting the k +1, ··· ,n columns and the k +1, ··· ,n rows from A. Thus A n = A and A k
is the k × k submatrix of A which occupies the upper left corner of A. The determinants of
these submatrices are called the principle minors.
The following theorem is proved in [8]
Theorem 13.4.6 Let X be a finite dimensional Hilbert space and let A ∈L(X, X) be self
adjoint. Then A is positive definite if and only if det (M (A) k
) > 0 for every k =1, ··· ,n.
Here M (A) denotes the matrix of A with respect to some fixed orthonormal basis of X.
i=1