wk2 Flashcards
What are matrices equivalent to
Linear Operators
How does linear operator composition work
AB|V> = A( B( |V> ) )
Suppose vector space V has basis vectors |0>, and |1>. If there is a projector A: V -> V. What is the the operator matrix
A = [ [1, 0], [0, 1
] ]
maybe
What makes two vectors orthogonal
Their inner product is 0 i.e. <v|w>=0
Define vector norm
|| |v> || = root ( <v|v> )
What is a unit vector
norm of vector = 1 i.e.
|| |v> || = 1
Define the orthonormal set |i> where i is the index
The set of vectors with index I is orthonormal if each vector is a unit vector and distinct vectors in the set are orthogonal to each other
How do you generate an orthonormal set
Gramm-Schmidtt Procedure then normalise vectors
How does Gramm-Schmidtt procedure work
First vector (v_1) is taken as a pivot point and hence first orthogonal vector (u_1). Then the following orthogonal vector u_2 = v_2 - projection_{u_1} (v_2)
Finally normalise the orthogonal u_n vectors
define the projection operator used in gramm schmidt
proj_u (v) = (<v, u> / <u,u>) u
What is the completeness relation
sum_i: |I><i|= Identity
Define EigenVector
A linear operator, A, on vector space is a non-zero vector |v> such that A|v> = V|v> where V is a complex number known as the eigenvalue of A corresponding to |v>
What is the characteristic function
C |lambda> = det|A - (lambda Identity) >
values C (lambda) = 0 are eigenvalues
what is a diagonalisable operator
an operator with diagonal representation such as [ [ 1, 0 ], [ 0, -1 ] ] = |0><0| - |1><1|
Sometimes known as the orthonormal decomposition
Define Hermitian operator
A^T = A
When an operators complex conjugate is equal to itself
