Understanding N in Jordan Block Matrix

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SUMMARY

The discussion centers on the Jordan block matrix, specifically defining the matrix N as having entries δi,j-1. This means that N has 1s directly above the diagonal and 0s elsewhere, leading to the conclusion that the diagonal entries of N are indeed all 0. The characteristic polynomial PJ(λ) of the Jordan block J, defined as J = λI + N, satisfies the equation PJ(J) = 0, confirming that J is an eigenvalue of itself.

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  • Understanding of Jordan block matrices
  • Familiarity with characteristic polynomials
  • Knowledge of matrix notation and operations
  • Basic linear algebra concepts
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  • Study the properties of Jordan forms in linear algebra
  • Learn about eigenvalues and eigenvectors in the context of matrices
  • Explore the derivation of characteristic polynomials for different matrix types
  • Investigate the implications of nilpotent matrices in linear transformations
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Wildcat
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Homework Statement



Let J be any Jordan block, i.e. J =λI + N where N is the matrix whose (i,j) entry is δi,j-1.
PJ(λ) is J's characteristic polynomial. Show that PJ(J)=0.

Homework Equations





The Attempt at a Solution



I don't understand what this part of the question means → N is the matrix whose (i,j) entry is δi,j-1. Can someone explain? Does it mean that for example the (2,2) entry of N would be the (2,1) entry of J which would be 0? Making the diagonal entries of N all =0??
 
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Wildcat said:

Homework Statement



Let J be any Jordan block, i.e. J =λI + N where N is the matrix whose (i,j) entry is δi,j-1.
PJ(λ) is J's characteristic polynomial. Show that PJ(J)=0.

Homework Equations





The Attempt at a Solution



I don't understand what this part of the question means → N is the matrix whose (i,j) entry is δi,j-1. Can someone explain? Does it mean that for example the (2,2) entry of N would be the (2,1) entry of J which would be 0? Making the diagonal entries of N all =0??


I don't know how to close threads, but while I was waiting for a reply i figured it out!
 

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