description/proof of that conjugate of Hermitian matrix by unitary matrix is Hermitian
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of Hermitian matrix.
- 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 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.
Target Context
- The reader will have a description and a proof of the proposition that the conjugate of any Hermitian matrix by any unitary matrix is Hermitian.
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:
\(M\): \(\in \{\text{ the Hermitian matrices }\}\)
\(N\): \(\in \{\text{ the unitary matrices }\}\)
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Statements:
\(N M N^{- 1} \in \{\text{ the Hermitian matrices }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(N M N^{- 1}\) is valid; Step 2: see that \((N M N^{- 1})^* = N M N^{- 1}\).
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, \(= N M N^{- 1}\), by 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.
So, \((N M N^{- 1})^*\) is Hermitian.
