description/proof of that for complex matrix, inverse of Hermitian conjugate of matrix is Hermitian conjugate of inverse of matrix
Topics
About: matrices space
The table of contents of this article
Starting Context
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.
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:
\(M\): \(\in \{\text{ the } m \times m \text{ invertible complex matrices }\}\)
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Statements:
\({M^*}^{-1} = {M^{-1}}^*\)
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2: Proof
Whole Strategy: Step 1: see that \({M^{-1}}^* M^* = I\) and \(M^* {M^{-1}}^* = I\).
Step 1:
\({M^{-1}}^* M^* = (M M^{-1})^*\), 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.
\(= I^* = I\).
\(M^* {M^{-1}}^* = (M^{-1} M)^*\), 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.
\(= I^* = I\).
That means that \({M^*}^{-1} = {M^{-1}}^*\).
