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ltrane2003
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Homework Statement
Prove that the product of the diagonal entries of an nxn matrix A equals the product of the eigenvalues of A.
ltrane2003 said:Homework Statement
Prove that the product of the diagonal entries of an nxn matrix A equals the product of the eigenvalues of A.
Dick said:But it's not even true!? Take A=[[0,1],[1,0]]. Product of the eigenvalues is -1. Product of the diagonal entries is 0. Am I missing something?
Vid said:Both the eigenvalues are zero.
For a diagonal matrix the determinant is just the product of the diagonals.
The eiganvalues are |xI-A|...
AstroRoyale said:The product of the eigenvalues is the determinant, right. The statement would be true for a diagonal matrix for sure :)
DavidWhitbeck said:Yeah I totally screwed up, the OP never returned but I'll just say what I was thinking-- the constant term in the characteristic polynomial is known to be the determinant of the matrix, call it A, but also if you factor the polynomial knowing that it's roots are the eigenvalues [tex]\lambda_i[/tex] then the constant term is also
[tex](-1)^N\prod_{i=1}^{N}\lambda_i[/tex]
and there you have it--
[tex]\prod_{i=1}^{N}\lambda_i = (-1)^N\det A[/tex]
where A is an N by N matrix.
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