Determinant of a Hermitiain matrix

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    Determinant Matrix
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Homework Help Overview

The discussion revolves around proving that the determinant of a Hermitian matrix is real. The original poster expresses uncertainty about inner products and seeks assistance in understanding the proof process.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the determinant of a matrix and its conjugate, with suggestions to prove that the determinant of the conjugate equals the conjugate of the determinant. There are inquiries about the specific formulas and processes involved in calculating the determinant.

Discussion Status

Some participants have offered guidance on the properties of Hermitian matrices and the determinants, while others are exploring the definitions and implications of these properties. There is a recognition of differing interpretations regarding the nature of formulas versus procedures in this context.

Contextual Notes

Participants note the original poster's lack of familiarity with inner products, which may affect their understanding of the concepts being discussed. There is also mention of the absence of a closed formula for determinants in general, which may contribute to the confusion.

kingwinner
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Q: Suppose A is a Hermitiain matrix, prove that det A is real.

Note: I know nothing about inner products yet.

Some thoughts:
Perhaps proving that det A = conjugate of (det A) ? But how?


Can someone please help me? Thanks!
 
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Try proving that det(conjugate of A) = conjugate of det(A) (use the formula for det(.)).
 
How can I do that? Can you please tell me more about it?
 
I'm not terribly sure as to what formula morphism is referring to. There is a process/algorithm for finding the determinant of a general matrix but no closed formula.

You know that by the definition of a Hermitian Matrix, it's equal to it's transpose conjugate. Furthermore, we know that the the determinant of the transpose is equal to the determinant of the original matrix. Thus using this bit of information, this is a one line proof.
 
Kreizhn said:
I'm not terribly sure as to what formula morphism is referring to. There is a process/algorithm for finding the determinant of a general matrix but no closed formula.

You know that by the definition of a Hermitian Matrix, it's equal to it's transpose conjugate. Furthermore, we know that the the determinant of the transpose is equal to the determinant of the original matrix. Thus using this bit of information, this is a one line proof.

But this just gives det A = det (conjugate of A)
 
The formula is here: http://planetmath.org/encyclopedia/Determinant2.html
 
Last edited by a moderator:
It may be considered a formula semantically in that following the outlined process will give you the determinant, but I don't believe that we can consider it a formula in the classical sense. Rather it's a procedure, which logically is a very different thing.

Furthermore, the determinant of the conjugate is the conjugate of the determinant.
 
Kreizhn said:
Furthermore, the determinant of the conjugate is the conjugate of the determinant.
Yes, and using that formula (or procedure - call it whatever you want) gives a very easy proof of that, like I stated in post #2.
 

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