description/proof of that complex conjugate of product of complex matrices is product of complex conjugates of constituents
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of complex conjugate of complex matrix.
Target Context
- The reader will have a description and a proof of the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents.
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:
\(M_1\): \(\in \{\text{ the } m \times n \text{ complex matrices }\}\)
\(M_2\): \(\in \{\text{ the } n \times o \text{ complex matrices }\}\)
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Statements:
\(\overline{M_1 M_2} = \overline{M_1} \text{ } \overline{M_2}\)
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2: Proof
Whole Strategy: Step 1: let \(M_1 = \begin{pmatrix} {M_1}^j_l \end{pmatrix}\) and \(M_2 = \begin{pmatrix} {M_2}^l_m \end{pmatrix}\); Step 2: see the components of \(\overline{M_1 M_2}\); Step 3: see the components of \(\overline{M_1} \text{ } \overline{M_2}\), and conclude the proposition.
Step 1:
Let \(M_1 = \begin{pmatrix} {M_1}^j_l \end{pmatrix}\) and \(M_2 = \begin{pmatrix} {M_2}^l_m \end{pmatrix}\).
Step 2:
\((M_1 M_2)^j_m = {M_1}^j_l {M_2}^l_m\).
\(\overline{M_1 M_2}^j_m = \overline{{M_1}^j_l {M_2}^l_m} = \overline{{M_1}^j_l} \text{ } \overline{{M_2}^l_m}\), because the conjugate of the product of any complex numbers is the product of the conjugates of the constituents.
Step 3:
\((\overline{M_1} \text{ } \overline{M_2})^j_m = \overline{{M_1}^j_l} \overline{{M_2}^l_m}\), \(= \overline{M_1 M_2}^j_m\), by the result of Step 2.
