- #1
Nusc
- 760
- 2
Homework Statement
Let S be the unilateral shift and compute SS^* and S^*S. Also compute S^n S^(*n) and S^(*n)S^n
Homework Equations
The Attempt at a Solution
First what is meant by unilateral shift?
Nusc said:First what is meant by unilateral shift?
In mathematics, a unilateral shift refers to a linear transformation that shifts or translates a set of points in one direction. In this case, it specifically refers to a shift in the direction of increasing powers of S.
SS* represents the conjugate transpose of the matrix S. This means that the rows and columns of S are interchanged and each element is replaced by its complex conjugate. In other words, it is the transpose of S with the imaginary part of each element flipped in sign.
S*S is calculated by multiplying the matrix S by its conjugate transpose, SS*.
S^nS^(*n) and S^(*n)S^n represent the product of S^n with its conjugate transpose and the product of the conjugate transpose of S^n with S^n, respectively. These products are important in computing the spectral norm of a matrix, which is a measure of the maximum amplification of a vector under the linear transformation represented by the matrix.
This equation is commonly used in linear algebra and functional analysis to study the properties of linear transformations and their effects on vectors and matrices. It can be applied in various fields such as physics, engineering, and computer science to model and analyze various systems. Additionally, it can be used to solve systems of linear equations and calculate eigenvalues and eigenvectors of a matrix.