Page 310 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes  It is shown that if A is the matrix of the form f in the ordered basis { , ... } of V and the
1 n
principal minors are all different from zero, then there exists a unique upper-triangular
matrix P with P =1 (1 k n) such that P*AP is upper triangular.
kk
28.4 Keywords

Non-negative Form: A form f on real or complex vector space V is non-negative if it is Hermitian
and f( , ) 0.

Positive Form: A form f is positive if it is Hermitian and f( , ) > 0
Upper Triangular Matrix: A matrix P is upper triangular one if its elements P satisfy the
ij
relations: P = 1, 1 k n and P = 0 for j > k.
kk ij
28.5 Review Questions

1. Let
1 1 2
A =
1 2 1 4
(a) Show that A is positive

(b) Find an invertible real matrix P such that
t
A = P P.
2. Does

,
,
(x x 2 )|(y y 2 ) x y 2x y 2x y x y 2
2 2 define an inner product on c ?
1
1 2
1 1
1
2 1
28.6 Further Readings
Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I N. Herstein, Topics in Algebra

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