L may not be 1, and you don't know what L is. But what is the length of Ux?Raven2816 said:hmmm, i know that, so i have Ux = Lx and L is 1...so then Ux = x...? I'm just getting lost
L is a number. There is a formula for pulling a number multiplier out of an inner product.Raven2816 said:and <Lx, Lx> is the inner product of an e-vector with its e-value...so do i get one? or am i using L=1?
What about <x,Ly>?Raven2816 said:i know that L<x, y> = <Lx, y> ...
A unitary operator is a mathematical concept used in linear algebra and functional analysis. It is a type of linear transformation that preserves the inner product between two vectors, meaning that the length and angle between the two vectors remains the same after the transformation.
The spectrum of a unitary operator refers to the set of all possible eigenvalues of the operator. When the spectrum is on the unit circle, it means that all eigenvalues have a magnitude of 1. This is significant because it implies that the operator is bijective, meaning it has both an inverse and a unique solution.
In quantum mechanics, unitary operators are used to describe the evolution of a quantum system over time. They represent the transformations that occur to a system as it interacts with its environment, and they preserve the probabilities of measurement outcomes.
Yes, unitary operators have applications in various fields such as signal processing, control theory, and computer science. In these fields, they are used for tasks such as filtering, error correction, and data compression.
Unitary operators have the unique property of preserving the norm of a vector, meaning that the length of a vector remains the same after the transformation. Other types of linear transformations, such as orthogonal operators, may change the length of a vector. Additionally, unitary operators are always invertible, while other types of transformations may not have an inverse.