1146: Hermitian Conjugate of Product of Complex Matrices Is Product of Hermitian Conjugates of Constituents in Reverse Order

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description/proof of that Hermitian conjugate of product of complex matrices is product of Hermitian conjugates of constituents in reverse order

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Topics

About: matrices space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

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Starting Context

• The reader knows a definition of Hermitian conjugate of complex matrix.
• The reader admits the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents.
• The reader admits the proposition that for any commutative ring, the transpose of the product of any matrices is the product of the transposes of the constituents in the reverse order.

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Target Context

• The reader will have a description and a proof of 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.

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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.

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Main Body

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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:
$(M_1{M}_2{)}^{\ast }={M_2}^{\ast }{M_1}^{\ast }$
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2: Proof

Whole Strategy: Step 1: to $(M_1{M}_2{)}^{\ast }$ apply the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents and the proposition that for any commutative ring, the transpose of the product of any matrices is the product of the transposes of the constituents in the reverse order, and see that it equals ${M_2}^{\ast }{M_1}^{\ast }$.

Step 1:

$(M_1{M}_2{)}^{\ast }={\overline{M_1{M}_2}}^t$.

$=(\overline{M_1}\overline{M_2}{)}^t$, by the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents.

$={\overline{M_2}}^t{\overline{M_1}}^t$, by the proposition that for any commutative ring, the transpose of the product of any matrices is the product of the transposes of the constituents in the reverse order.

$={M_2}^{\ast }{M_1}^{\ast }$.

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References

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