Unilateral Shift Homework: Compute SS*, S*S, S^nS^(*n), S^(*n)S^n

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In summary, a unilateral shift in mathematics refers to a linear transformation that shifts a set of points in one direction. In this context, it specifically refers to a shift in the direction of increasing powers of S. SS* in the equation represents the conjugate transpose of the matrix S, which is calculated by interchanging rows and columns and flipping the imaginary part of each element in sign. S*S is calculated by multiplying S with its conjugate transpose. The products of S^nS^(*n) and S^(*n)S^n are important in computing the spectral norm of a matrix, which is used in various fields to study linear transformations. This equation can also be applied in research and experiments in fields such as physics, engineering, and
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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?
 
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Related to Unilateral Shift Homework: Compute SS*, S*S, S^nS^(*n), S^(*n)S^n

1. What is the meaning of "unilateral shift" in this context?

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.

2. What does SS* represent in this equation?

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.

3. How is S*S calculated?

S*S is calculated by multiplying the matrix S by its conjugate transpose, SS*.

4. What is the significance of S^nS^(*n) and S^(*n)S^n in this equation?

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.

5. How can I use this equation in my research or experiments?

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.

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