1149: For Complex Matrix, Inverse of Hermitian Conjugate of Matrix Is Hermitian Conjugate of Inverse of Matrix

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description/proof of that for complex matrix, inverse of Hermitian conjugate of matrix is Hermitian 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 Hermitian conjugate of complex matrix.
• The reader admits 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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Target Context

• The reader will have a description and a proof of the proposition that for any invertible complex matrix, the inverse of the Hermitian conjugate of the matrix is the Hermitian 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:
${M^{\ast }}^{-1}={M^{-1}}^{\ast }$
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2: Proof

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

Step 1:

${M^{-1}}^{\ast }{M}^{\ast }=(M{M}^{-1}{)}^{\ast }$, by 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.

$=I^{\ast }=I$.

$M^{\ast }{M^{-1}}^{\ast }=(M^{-1}M{)}^{\ast }$, by 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.

$=I^{\ast }=I$.

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

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References

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