description/proof of that inverse 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 admits the proposition that the Hermitian conjugate of any unitary matrix is unitary.
Target Context
- The reader will have a description and a proof of the proposition that the inverse 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:
\(M\): \(\in \{\text{ the unitary matrices }\}\)
\(M^{-1}\): \(= \text{ the inverse of } M\)
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Statements:
\(M^{-1} \in \{\text{ the unitary matrices }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(M^{-1} = M^*\); Step 2: apply the proposition that the Hermitian conjugate of any unitary matrix is unitary.
Step 1:
\(M^{-1} = M^*\), by the definition of unitary matrix.
Step 2:
\(M^*\) is unitary, by the proposition that the Hermitian conjugate of any unitary matrix is unitary, so, \(M^{-1}\) is unitary.
