Finding Eigenvalues of an Arbitrary Matrix

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    Eigenvalues Matrix
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Discussion Overview

The discussion revolves around finding the eigenvalues of the matrix C = A - αI, where A is an arbitrary matrix and I is the identity matrix. Participants explore the implications of eigenvalues in relation to the original matrix A and the modified matrix C.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to find the eigenvalues of C = A - αI.
  • Another participant seeks clarification on whether the question pertains to finding eigenvalues of A or specifically for a given α.
  • A participant states that the eigenvalues of any square matrix M are the roots of the polynomial det(M - xI) and notes that knowing the eigenvalues of A allows one to determine those of C.
  • One participant expresses uncertainty about finding the eigenvalues of the specific matrix C and states the need to prove that λ is an eigenvalue of A if and only if (λ - α) is an eigenvalue of C.
  • Another participant provides a condition for eigenvalues, stating that t is an eigenvalue of M if and only if M - tI is not invertible, and suggests filling in the blanks regarding the relationship between C and A.
  • A later reply indicates understanding of the discussion after receiving clarification.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method for finding eigenvalues, and there are competing views on the interpretation of the original question. The discussion remains unresolved regarding the specific steps needed to prove the relationship between the eigenvalues of A and C.

Contextual Notes

The discussion includes assumptions about the properties of eigenvalues and the invertibility of matrices, which may depend on the specific characteristics of matrix A and the scalar α.

himurakenshin
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How can i find the eigen value(s) of A - (alpha)I
where A is an arbitrary matrix ?
 
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Your question is ambiguous. Do you mean just find the eigenvalues of A- which would mean solving the equation det(A- alpha*I)= 0 for alpha or do you mean specifically finding eigenvalues of A- alpha*I for a given value of I?
 
sorry, where I is the identity matrix.
the matrix is C=(A-alpha*I)
I need to find the eigen values of C
 
the eigenvalues of any square matrix, call it M, are the roots of the polynomial in x

det(M-xI)

although if you know the eigen values of A you know them of C too.
 
matt grime said:
the eigenvalues of any square matrix, call it M, are the roots of the polynomial in x

det(M-xI)
yes I know this, but I don't know how to find the eigen value of that paticular matrix (A can be any matrix). The actual question is that I have to prove that lambda is an eigen value of A only if (lamda - alpha) is an eigen value of C
 
well, that wasn't what you asked was it?

t is an eigenvalue of M if and only if M-tI is not invertible.

let a be alpha
If C-tI=A-aI-tI is not invertible, then A-(a+t)I is not invertible, can you fill in the blanks?
 
got it. thanks a lot :)
 

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