Determinant of a Unitary Matrix

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Homework Help Overview

The discussion revolves around the properties of unitary matrices, specifically focusing on the relationship between the magnitude of a vector and its transformation under a unitary matrix. The original poster attempts to show that the magnitude of the vector remains unchanged when multiplied by a unitary matrix.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the properties of unitary matrices, questioning the interpretation of the notation used for magnitude versus determinant. There are attempts to relate the determinant of the unitary matrix to the magnitudes of vectors.

Discussion Status

The discussion is ongoing, with some participants providing insights into the properties of unitary matrices and questioning the notation used. There is a recognition that the original interpretation may need reevaluation, particularly regarding the distinction between magnitude and determinant.

Contextual Notes

There is confusion regarding the notation used for magnitude and determinant, which may affect the understanding of the problem. Participants are also considering the implications of the properties of unitary matrices in the context of the problem statement.

thepopasmurf
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Homework Statement


Let U be a unitary matrix. Show that for all vectors x that
|Ux| = |x|

Homework Equations



[tex]U^H=U^{-1}[/tex]
[tex]|Ux|=|U||x|[/tex]

The Attempt at a Solution



[tex]U^HU=I[/tex]
[tex]|U^HU|=1[/tex]
[tex]|U^T|^*|U|=1[/tex]
[tex](det(U))^2 = 1[/tex]
so
det(U) = +/- 1

But that doesn't solve the question
 
Last edited:
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First, convince yourself that [itex]|a|=aa^H[/itex]. Then use this to calculate |Ux| and |x|.
 
Hang on, the vertical bars are standing for the magnitude, not the determinant? I need to do this all again.
 
Uh, well, if they mean determinant, then I have no idea what |x| would mean. You can't take the determinant of a vector...
 

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