Page 306 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes positive forms? If f is a form on a complex vector space and A is the matrix of f in some ordered
basis, then f will be positive if and only if A = A* and
X*AX > 0 for all complex X 0 ....(8)

It follows from Theorem 3 of unit 27 that the condition A = A* is redundant, i.e., that (8) implies
A = A*. One the other hand, if we are dealing with a real vector space the form f will be positive
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if and only if A = A and
X*AX > 0 for all real X 0 ....(9)
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We want to emphasize that if a real matrix A satisfies (9), it does not follow that A = A . One thing
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which is true is that, if A = A and (9) holds, then (8) holds as well. That is because
(X + iY)*A(X + iY) = (X – iY )A(X + iY)
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t
t
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= X AX + Y AY + i[X AY – Y AX]
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t
t
t
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and if A = A then Y AX = X AY.
If A is an n n matrix with complex entries and if A satisfies (9), we shall call A a positive matrix.
Now suppose that V is a finite-dimensional inner product space. Let f be a non-negative form on
V. There is a unique self-adjoint linear operator T on V such that
f( , ) = (T | ) ...(10)
and T has the additional property that (T | ) 0
Definition: A linear operator T on a finite-dimensional inner product space V is non-negative if
T = T* and (T | ) 0 for all in V. A positive linear operator is one such that T = T* and
(T | ) > 0 for all 0.
If V is a finite-dimensional (real or complex) vector space and if (.|.) is an inner product on V,
there is an associated class of positive linear operators on V. Via (10) there is a one-one
correspondence between that class of positive operators and the collection of all positive forms
on V. Let us summarise as:
If A is an n n matrix over the field of complex numbers, the following are equivalent:

1. A is positive, i.e. A x x 0 whenever x , ..., x are complex numbers, not all 0.
kj j k
1
n
j k
2. (X|Y) = Y*AX is an inner product on the space of n 1 complex matrices.
3. Relative to the standard inner product (X|Y) = Y*X on n 1 matrices, the linear operator
X AX is positive.
4. A = P*P for some invertible n n matrix P over C.

5. A = A*, and the principal minors of A are positive.
If each entry of A is real, these are equivalent to:

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1. A = A , and A x x 0 whenever x , ..., x are real numbers, not all 0.
kj j k
1
n
j k
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2. (X|Y) = Y AX is an inner product on the space of n 1 real matrices.
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3. Relative to the standard inner product (X|Y) = Y X on n 1 real matrices, the linear
operator X AX is positive.
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4. There is an invertible n n matrix P, with real entries, such that A = P P.

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