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What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them?

So if I calculated determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors, what are ways to be sure that I didn't do a major mistake? I don't want to verify my solutions all the way through, I just want a quick way which gives me that it is highly likely that the calculated determinant is right etc.


Let $A$ be a matrix $A \in \operatorname{Mat}(n, \mathbb{C})$,

let $\det(A)$ be the determinant of matrix $A$,

let $v_1, v_2, ..., v_k$ be eigenvectors of matrix $A$,

let $\lambda_1, \lambda_2, ..., \lambda_n$ be eigenvalues of matrix $A$,

let $\chi_A(t) = t^n + a_{n-1}t^{n-1}+\cdots + a_0 = (t-\lambda_1)\cdots(t-\lambda_n)$ be the characteristic polynomial of matrix $A$,

let $\mu_A(t)$ be the minimal polynomial of matrix $A$.


Verifications suggested so far:

eigenvectors / eigenvalues

  • $\det(A) = \lambda_1^{m_1} \lambda_2^{m_2} \cdots \lambda_n^{m_l}$ where $m_i$ is the multiplicity of the corresponding eigenvalue
  • $a_0 = (-1)^n\lambda_1\cdots\lambda_n$
  • eigenvectors can be verified by multiplying with the matrix; the eigenvalues can be verified at the same time; i.e. $A v_i = \lambda_i v_i$

determinant

  • $\det(A) = \lambda_1^{m_1} \lambda_2^{m_2} \cdots \lambda_l^{m_l}$ where $m_i$ is the multiplicity of the corresponding eigenvalue

characteristic / minimal polynomial

  • $a_0 = (-1)^n\lambda_1\cdots\lambda_n$
  • $\mu_A(A) = 0$ and $\chi_A(A) = 0$
  • $\mu_A(t) \mid \chi_A(t)$
  • 0
    One more trick: eigenvectors $u, v$ have different eigenvalues $\mu \neq \lambda$ are linear independent. Note that you can use the fact that $u$ and $v$ are linear independent iff $n \times 2$ matrix $[u, v]$ has rank 2.2013-12-22

1 Answers 1

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You can use the Caley-Hamilton theorem. Put your matrix in the characteristic polynomial and you should find $0$. Explicitly, $\chi_A(A) = A^n + a_{n-1}A^{n-1}+\cdots + a_0I=0$. This checks the
characteristic polynomial, the minimal polynomial and the eigenvalues. The only thing not checked is the dimension of the eigenspace, and the eigenvectors.

Of course this checking is simple, if your software computes matrix algebra.

  • 2
    Note that the wrong characteristic polynomial can pass this test; e.g. if the minimal polynomial of a $4 \times 4$ matrix is $x(x-1)$, then plugging $A$ into any of $x^3 (x-1)$, $x^2 (x-1)^2$, and $x(x-1)^3$, or even $x(x-1)(x-2)(x-3)$ will give zero.2014-04-07