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