Proving a Theorem, related to Gerschgorin

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Homework Help Overview

The discussion revolves around proving a theorem related to Gerschgorin, specifically concerning the spectral radius of a complex matrix. The original poster presents a theorem involving a matrix \( A \) and its spectral radius \( p(A) \), seeking assistance in the proof process.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the need for definitions related to the theorem, such as the meaning of \( C_{n \times n} \) and \( p(A) \). There is an attempt to clarify these terms, particularly the nature of the spectral radius and its geometric interpretation. One participant suggests starting with a simpler case where the matrix is diagonalizable.

Discussion Status

The discussion is ongoing, with participants providing clarifications and exploring foundational concepts. There is no explicit consensus yet, but the conversation is moving towards understanding the definitions and potential starting points for the proof.

Contextual Notes

Some participants express uncertainty about the definitions and notation used in the theorem, indicating a need for clearer foundational knowledge before proceeding with the proof.

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Homework Statement


Here is the theorem I need to prove:

For A=(aij)\inCnxn

we have

p(A)\leqmax_{i}\Sigma^{n}_{j=1}|aij|


Homework Equations





The Attempt at a Solution


I have no idea how to go about this. :cry:
 
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Some definitions might be nice.
What is C_{n \times n}? What is p(A)? What does i run over?
 
Sorry,
Cnxn is the set of nxn matrices with entries in the complex number system.

p(A)= max{|\lambda1|, |\lambda2|, ...}

p(A) is the smallest circle in the comple plane, centered at the origin, which contains all the characteristic values of A.
 
Maybe it's a good idea to start with the simple case, where A is diagonalizable.
 

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