description/proof of that Hermitian conjugate of unitary matrix is unitary
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of unitary matrix.
- The reader knows a definition of Hermitian conjugate of complex matrix.
- The reader admits the proposition that the Hermitian conjugate of the Hermitian conjugate of any complex matrix is the matrix.
Target Context
- The reader will have a description and a proof of the proposition that the Hermitian conjugate of any unitary matrix is unitary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the unitary matrices }\}\)
\(M^*\): \(= \text{ the Hermitian conjugate of } M\)
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Statements:
\(M^* \in \{\text{ the unitary matrices }\}\)
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2: Proof
Whole Strategy: Step 1: see that \({M^*}^* = M\); Step 2: see that \({M^*}^* = {M^*}^{-1}\).
Step 1:
\({M^*}^* = M\), by the proposition that the Hermitian conjugate of the Hermitian conjugate of any complex matrix is the matrix.
Step 2:
\({M^*}^* M^* = M M^* = I\), because \(M\) is unitary.
\(M^* {M^*}^* = M^* M = I\), because \(M\) is unitary.
So, \({M^*}^* = {M^*}^{-1}\).
So, \(M^*\) is unitary.
