Find out if nxn matrix is diagonalisable based on det(XI-A)

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Homework Help Overview

The discussion revolves around determining the diagonalizability of an nxn matrix, A, particularly through the evaluation of the determinant of (XI - A). Participants explore the implications of eigenvalues and eigenvectors in this context.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants attempt to establish a connection between the determinant and the distinctness of eigenvalues as a criterion for diagonalizability. Others question the sufficiency of distinct eigenvalues and raise the issue of matrices like the identity matrix, which, despite having repeated eigenvalues, are still considered diagonalisable.

Discussion Status

The discussion is active, with participants exploring different interpretations of diagonalizability. Some guidance has been offered regarding the relationship between eigenvalues and eigenvectors, but no consensus has been reached on the criteria for diagonalizability.

Contextual Notes

There is an ongoing examination of the definitions and assumptions related to diagonalizability, particularly concerning the role of distinct eigenvalues and the conditions under which a matrix may still be diagonalisable.

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Homework Statement



If you need to check whether an nxn matrix, A, is diagonalisable or not, do you just find out what det(XI-A) is, and then if X has n distinct values it is diagonalisable, otherwise it's not.



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The Attempt at a Solution

 
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Consider the identity matrix...
 


The identity matrix has only one distinct solution, 1, but it is diagonalisable...

So how do you go about checking whether or not a matrix is diagonalisable or not?
 


A nxn square matrix is diagonizable if it has n linearly independent eigenvectors. Having n distinct eigenvalues is sufficient but not necessary for diagonalizability.
 

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