Lecture 38: Unitary operators
... that T is an isomorphism of inner product spaces. In this case T −1 also preserves inner products. A co-ordinate system C : Fn −→ V is said to be orthonormal if C is an isomorphism of inner product spaces. Proof. Suppose T preserves inner products & v ∈ ker T . Then 0 = (T v|T v) = (v|v) so v = 0 wh ...
... that T is an isomorphism of inner product spaces. In this case T −1 also preserves inner products. A co-ordinate system C : Fn −→ V is said to be orthonormal if C is an isomorphism of inner product spaces. Proof. Suppose T preserves inner products & v ∈ ker T . Then 0 = (T v|T v) = (v|v) so v = 0 wh ...
Formal power series
... and let b_n = number of domino tilings of a 3-by-(2n+1) rectangle with a bite taken out of one corner. a_n = 2b_{n-1} + a_{n-1} b_n = a_n+b_{n-1} = 3b_{n-1} + a_{n-1}. Initial values: a_0 = 1, a_1 = 3, b_0 = 1, b_1 = 4. Generating function approach: A la Wilf. “Transfer matrix approach”: Write a_n = ...
... and let b_n = number of domino tilings of a 3-by-(2n+1) rectangle with a bite taken out of one corner. a_n = 2b_{n-1} + a_{n-1} b_n = a_n+b_{n-1} = 3b_{n-1} + a_{n-1}. Initial values: a_0 = 1, a_1 = 3, b_0 = 1, b_1 = 4. Generating function approach: A la Wilf. “Transfer matrix approach”: Write a_n = ...
Construction of Transition Matrices for Reversible Markov Chains
... Author’s Declaration of Originality I hereby certify that I am the sole author of this major paper and that no part of this major paper has been published or submitted for publication. I certify that, to the best of my knowledge, my major paper does not infringe upon anyone’s copyright nor violate a ...
... Author’s Declaration of Originality I hereby certify that I am the sole author of this major paper and that no part of this major paper has been published or submitted for publication. I certify that, to the best of my knowledge, my major paper does not infringe upon anyone’s copyright nor violate a ...
ch7
... We obtain the transpose of a matrix by writing its rows as columns (or equivalently its columns as rows). This also applies to the transpose of vectors. Thus, a row vector becomes a column vector and vice versa. In addition, for square matrices, we can also “reflect” the elements along the main diag ...
... We obtain the transpose of a matrix by writing its rows as columns (or equivalently its columns as rows). This also applies to the transpose of vectors. Thus, a row vector becomes a column vector and vice versa. In addition, for square matrices, we can also “reflect” the elements along the main diag ...
Vector Norms
... In computing the solution to any mathematical problem, there are many sources of error that can impair the accuracy of the computed solution. The study of these sources of error is called error analysis, which will be discussed later in this lecture. First, we will focus on one type of error that oc ...
... In computing the solution to any mathematical problem, there are many sources of error that can impair the accuracy of the computed solution. The study of these sources of error is called error analysis, which will be discussed later in this lecture. First, we will focus on one type of error that oc ...
