1147: For Complex Matrix, Inverse of Complex Conjugate of Matrix Is Complex Conjugate of Inverse of Matrix

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description/proof of that for complex matrix, inverse of complex conjugate of matrix is complex conjugate of inverse of matrix

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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 complex 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.

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Target Context

• The reader will have a description and a proof of the proposition that for any invertible complex matrix, the inverse of the complex conjugate of the matrix is the complex conjugate of the inverse of the matrix.

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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:
$M$: $\in \{\text{ the }m\times m\text{ invertible complex matrices }\}$
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Statements:
${\bar{M}}^{-1}=\overline{M^{-1}}$
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2: Proof

Whole Strategy: Step 1: see that $\overline{M^{-1}}\bar{M}=I$ and $\bar{M}\overline{M^{-1}}=I$.

Step 1:

$\overline{M^{-1}}\bar{M}=\overline{M^{-1}M}$, by the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents, $=\bar{I}=I$.

$\bar{M}\overline{M^{-1}}=\overline{M{M}^{-1}}$, by the proposition that the complex conjugate of the product of any complex matrices is the product of the complex conjugates of the constituents, $=\bar{I}=I$.

That means that $\overline{M^{-1}}={\bar{M}}^{-1}$.

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References

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