How to compute the characteristic equation given the eigenvalues?

Is there a matlab command to get the characteristic equation (CE) given the eigenvalues? Similarly how to get CE from the A matrix?

Answer 1

[John D'Errico]

You have not asked this question as if it is a homework assignment, so I'll give an answer.

The characteristic equation GIVEN eigenvalues? Trivial.

The eigenvalues are the roots of a polynomial, known as the characteristic polynomial. What polynomial has roots at a list of points? For example, suppose the eigenvalues are:

EVList = randn(1,5)

EVList = 1×5

-0.4199 -1.3665 1.3929 -1.3947 -0.3760

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What polynomial has those roots?

syms lambda
CP = prod(lambda - EVList)

CP = 

Feel free to expand it. I'll show only 5 digits, to make it readable.

vpa(expand(CP),5)

ans = 

You could recover the roots to convince yourself that it worked, but I'll leave that to you.

Can you learn the eigenvalues from a matrix? (HInt: what does eig do?)

What is the characteristic polynomial, given a matrix A?

A = randn(3,3)

A = 3×3

-1.0183 0.6000 -0.1730 1.2929 0.3481 0.4041 -0.1120 -1.2856 1.1605

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CP2 = det(A - lambda*eye(size(A)))

CP2 = 

Can we find the roots of that polynomial? You can use solve, or you could extract the coefficients, and then use roots.

solve(CP2)

ans = 

Are they the same as the eigenvalues, as given by eig?

eig(A)

ans = 
  -1.3743 + 0.0000i
   0.9323 + 0.5344i
   0.9323 - 0.5344i