[math-fun] Characteristic polynomial question
What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Since the characteristic polynomial is monic, perhaps you mean the "monicized" derivative. Victor On Thu, Oct 15, 2015 at 12:36 PM, Mike Stay <metaweta@gmail.com> wrote:
What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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On Thu, Oct 15, 2015 at 10:06 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Since the characteristic polynomial is monic, perhaps you mean the "monicized" derivative.
I'd forgotten that fact. But if there's one that uses the monicized derivative, that would be interesting, too. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Solve for the roots of the derivative polynomial. These are the eigenvalues of M'. String them along the diagonal, and make an arbitrary similarity transformation. If M' has repeated eigenvalues, Jordan blocks are also possible. -- Gene From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 15, 2015 9:36 AM Subject: [math-fun] Characteristic polynomial question What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Mike, surely the relation you're looking for is the fact that the derivative of the characteristic polynomial of M is the sum of the characteristic polynomials of its principal submatrices M_1, M_2,..., M_n, where M_i is what you get by dropping the ith row and column from M. Oh hey look, proof here: http://math.stackexchange.com/questions/978815/is-the-derivative-of-the-char... --Michael On Thu, Oct 15, 2015 at 2:59 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
Solve for the roots of the derivative polynomial. These are the eigenvalues of M'. String them along the diagonal, and make an arbitrary similarity transformation. If M' has repeated eigenvalues, Jordan blocks are also possible.
-- Gene
From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 15, 2015 9:36 AM Subject: [math-fun] Characteristic polynomial question
What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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-- Forewarned is worth an octopus in the bush.
Excellent, thank you! On Thu, Oct 15, 2015 at 12:22 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
Mike, surely the relation you're looking for is the fact that the derivative of the characteristic polynomial of M is the sum of the characteristic polynomials of its principal submatrices M_1, M_2,..., M_n, where M_i is what you get by dropping the ith row and column from M.
Oh hey look, proof here: http://math.stackexchange.com/questions/978815/is-the-derivative-of-the-char...
--Michael
On Thu, Oct 15, 2015 at 2:59 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
Solve for the roots of the derivative polynomial. These are the eigenvalues of M'. String them along the diagonal, and make an arbitrary similarity transformation. If M' has repeated eigenvalues, Jordan blocks are also possible.
-- Gene
From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 15, 2015 9:36 AM Subject: [math-fun] Characteristic polynomial question
What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Gene's answer suggests the question of whether there exists a continuous mapping f: M_n[C] —> M_(n-1)[C] from n x n matrices over the complexes C to (n-1)x(n-1) matrices over C such that in C[t], (d/dt)charpoly(M)(t) = charpoly(f(M))(t) for all M in M_n[C]. —Dan
On Oct 15, 2015, at 12:26 PM, Mike Stay <metaweta@gmail.com> wrote:
On Thu, Oct 15, 2015 at 2:59 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com>> wrote:
Solve for the roots of the derivative polynomial. These are the eigenvalues of M'. String them along the diagonal, and make an arbitrary similarity transformation. If M' has repeated eigenvalues, Jordan blocks are also possible.
From: Mike Stay <metaweta@gmail.com <mailto:metaweta@gmail.com>> To: math-fun <math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com>> Sent: Thursday, October 15, 2015 9:36 AM Subject: [math-fun] Characteristic polynomial question
What operation on a matrix M gives a new matrix M' whose characteristic polynomial is the derivative of the char. poly. of M?
participants (5)
-
Dan Asimov -
Eugene Salamin -
Michael Kleber -
Mike Stay -
Victor Miller