Mathematical Physics Vol 1

1.6 Gram-Schmidt orthogonalization procedure

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The numbers A ji are called components of linear operator A (or vector function A ) in the coordinat system e i . Specifically, it follows from (1.66) that A ji is the j –th component of the vector A e i . Similar as vectors, linear operators often have their physical meaning, which is independent of a specific coordinate system, and can be described without a coordinate system. This can be expressed symbolically as follows. The addition and multiplication of linear operators and multiplication of a linear operator bya scalar can be defined by the following relations ( A + B ) x = A x + B x (1.70) ( AB ) x = A ( B x ) (1.71) ( λ A ) x = λ ( A x ) , (1.72) for each x ∈ X , where X represents a vector space. In the general case AB̸ = BA . If the products are equal, then it is said that the "multiplication" is commutative. Let is define the zero ( 0 ) operator and the identity operator ( I ) by the following relations 0 x = 0 and I x = x , (1.73) for each x from the observed space. Two operators are equal if the following is true A x = B x (1.74) for each vector x ∈ X . Finally, if there exists an operator A − 1 with the following property AA − 1 = A − 1 A = I , (1.75) than this operator A − 1 is called inverse operator for operator A . Operators that have inverse operators are called nonsingular . 1.6 Gram-Schmidt orthogonalization procedure

Theorem1 If the non-zero vectors u 1 ,..., u k ∈ R { u 1 ,..., u k } is linearly dependent.

k are mutually orthogonal, then the set of vectors

Proof

c 1 u 1 + ··· + c k u k = 0 |· u i c i ( u i · u i )= 0 ⇒ c i = 0 , because u i is not a zero-vector, i.e. u i · u i > 0, the other members are equal to zero due to the orthogonality among vectors.