1145: Complex Conjugate of Product of Complex Matrices Is Product of Complex Conjugates of Constituents|T.B.P.

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description/proof of that complex conjugate of product of complex matrices is product of complex conjugates of constituents

Topics

About: matrices space

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.

M_{1}

\in {the m \times n complex matrices}

M_{2}

\in {the n \times o complex matrices}

//

Statements:

\overset{―}{M_{1} M_{2}} = \overset{―}{M_{1}} \overset{―}{M_{2}}

Whole Strategy: Step 1: let

M_{1} = (\begin{matrix} {M_{1}}_{l}^{j} \end{matrix})

and

M_{2} = (\begin{matrix} {M_{2}}_{m}^{l} \end{matrix})

; Step 2: see the components of

\overset{―}{M_{1} M_{2}}

; Step 3: see the components of

\overset{―}{M_{1}} \overset{―}{M_{2}}

, and conclude the proposition.

Step 1:

Let

M_{1} = (\begin{matrix} {M_{1}}_{l}^{j} \end{matrix})

and

M_{2} = (\begin{matrix} {M_{2}}_{m}^{l} \end{matrix})

.

Step 2:

(M_{1} M_{2})_{m}^{j} = {M_{1}}_{l}^{j} {M_{2}}_{m}^{l}

{\overset{―}{M_{1} M_{2}}}_{m}^{j} = \overset{―}{{M_{1}}_{l}^{j} {M_{2}}_{m}^{l}} = \overset{―}{{M_{1}}_{l}^{j}} \overset{―}{{M_{2}}_{m}^{l}}

, because the conjugate of the product of any complex numbers is the product of the conjugates of the constituents.

Step 3:

(\overset{―}{M_{1}} \overset{―}{M_{2}})_{m}^{j} = \overset{―}{{M_{1}}_{l}^{j}} \overset{―}{{M_{2}}_{m}^{l}}

= {\overset{―}{M_{1} M_{2}}}_{m}^{j}

, by the result of Step 2.

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