description/proof of that for unitary matrix, Hermitian conjugate of inverse of matrix and inverse of Hermitian conjugate of matrix is matrix
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 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 for any unitary matrix, the Hermitian conjugate of the inverse of the matrix and the inverse of the Hermitian conjugate of the matrix is the matrix.
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:
\(M\): \(\in \{\text{ the unitary matrices }\}\)
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Statements:
\({M^{- 1}}^* = {M^*}^{- 1} = M\)
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2: Proof
Whole Strategy: Step 1: see that \({M^{- 1}}^* = {M^*}^{- 1} = M\).
Step 1:
\({M^{- 1}}^* = {M^*}^* = M\), by the proposition that the Hermitian conjugate of the Hermitian conjugate of any complex matrix is the matrix.
\({M^*}^{- 1} = {M^{- 1}}^{- 1} = M\).
