Can I Write a Hermitian Matrix as A=A^(1/2)AH^(1/2)?

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Discussion Overview

The discussion revolves around the representation of a Hermitian matrix A in the form A = A^(1/2)A^(H/2), where H denotes the Hermitian operation. Participants explore the implications of this representation and clarify the notation used.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant asks if a Hermitian matrix A can be expressed as A = A^(1/2)A^(H/2).
  • Another participant seeks clarification on the notation H/2, suggesting it might refer to (A^H)^(1/2).
  • A participant confirms that H/2 indeed means the square root of the complex conjugate transpose of A.
  • It is noted that since A is Hermitian, A^H = A, allowing the expression to be simplified to A = A^(1/2)A^(1/2).
  • One participant acknowledges the clarification and thanks another for the insight.

Areas of Agreement / Disagreement

Participants generally agree on the simplification of the expression due to the properties of Hermitian matrices, but there is initial confusion regarding the notation used.

Contextual Notes

The discussion does not resolve potential limitations in the interpretation of the notation or the broader implications of the representation.

EngWiPy
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Hello,

If I have a Hermitian matrix A, can I write it as:

\mathbf{A}=\mathbf{A}^{1/2}\mathbf{A}^{H/2}

where superscript H denotes hermitian operation?

Thanks
 
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What do you mean with ##H/2##? Do you mean

(A^H)^{1/2}

or something similar?
 
micromass said:
What do you mean with ##H/2##? Do you mean

(A^H)^{1/2}

or something similar?

Yes exactly, it means the square root of the complex conjugation transpose of A.
 
OK, but if ##A## is hermitian, then ##A^H = A##, no? So you can write your original post as

A = A^{1/2}A^{1/2}

which is certainly true.
 
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micromass said:
OK, but if ##A## is hermitian, then ##A^H = A##, no? So you can write your original post as

A = A^{1/2}A^{1/2}

which is certainly true.

Oh, I didn't see it that way. Thanks
 

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