description/proof of that conjugate of unitary matrix by 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 conjugate of matrix by invertible matrix.
- The reader admits the proposition that for any ring, the multiplications of any matrices over the ring are associative.
- The reader admits the proposition that the Hermitian conjugate of the product of any complex matrices is the product of the Hermitian conjugates of the constituents in the reverse order.
- The reader admits the proposition that the inverse of any unitary matrix is unitary.
Target Context
- The reader will have a description and a proof of the proposition that the conjugate of any unitary matrix by 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:
\(M\): \(\in \{\text{ the unitary matrices }\}\)
\(N\): \(\in \{\text{ the unitary matrices }\}\)
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Statements:
\(N M N^{- 1} \in \{\text{ the unitary matrices }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(N M N^{- 1}\) is valid; Step 2: see that \((N M N^{- 1})^* = (N M N^{- 1})^{- 1}\).
Step 1:
\(N\) has \(N^{- 1}\), because \(N^{- 1} = N^*\), by the definition of unitary matrix.
So, \(N M N^{- 1}\) is valid.
Step 2:
We hereafter use the proposition that for any ring, the multiplications of any matrices over the ring are associative.
\((N M N^{- 1})^* = {N^{- 1}}^* M^* N^*\), by the proposition that the Hermitian conjugate of the product of any complex matrices is the product of the Hermitian conjugates of the constituents in the reverse order.
So, \((N M N^{- 1})^* N M N^{- 1} = {N^{- 1}}^* M^* N^* N M N^{- 1} = {N^{- 1}}^* M^* (N^* N) M N^{- 1} = {N^{- 1}}^* M^* I M N^{- 1} = {N^{- 1}}^* (M^* M) N^{- 1} = {N^{- 1}}^* I N^{- 1} = {N^{- 1}}^* N^{- 1} = I\), by the proposition that the inverse of any unitary matrix is unitary.
Likewise, \(N M N^{- 1} (N M N^{- 1})^* = N M N^{- 1} {N^{- 1}}^* M^* N^* = N M (N^{- 1} {N^{- 1}}^*) M^* N^* = N M I M^* N^*\), by the proposition that the inverse of any unitary matrix is unitary, \(= N (M M^*) N^* = N I N^* = N N^* = I\).
So, \((N M N^{- 1})^* = (N M N^{- 1})^{- 1}\).
So, \((N M N^{- 1})^*\) is unitary.
