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Is it true that for a square matrix $A$, all of whose eigenvalues exist in the base field, sum of the eigenvalues = trace($A$)?

The result holds in all the matrices I've studied.

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    $\begingroup$ Being in the base field is not necessary. $\endgroup$ Commented May 25, 2013 at 5:27
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    $\begingroup$ @tessellation "the coefficient of the linear term of any monic polynomial is the sum of its roots with a minus sign" $$(x-3)(x-2)(x-1)=x^3-6x^2+11x-6$$ $$3+2+1\ne -11$$ You should be looking at the second highest degree term, if you are interested in the sum of its roots. $\endgroup$ Commented May 25, 2013 at 5:47

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Yes. Just look at the characteristic polynomial (say of degree n). Trace=-the coefficient of the term of $x^{(n-1)}$ which is also the sum of the roots of the characteristic polynomial (the coefficient of the term $x^{(n-1)}$ of any monic polynomial of degree $n$ is the sum of its roots with a minus sign.).

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  • $\begingroup$ Exactly. And the same reasoning shows why the determinant is the product of the eigenvalues. $\endgroup$
    – awwalker
    Commented May 25, 2013 at 5:26
  • $\begingroup$ @Ethan thanks for pointing out the mistake. Actually I was thinking in terms of monic polynomials. Thanks again. $\endgroup$ Commented May 25, 2013 at 5:36
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    $\begingroup$ @EwanDelanoy No, that isn't true either, $$(x-3)(x-2)(x-1)=x^3-6x^2+11x-6$$ $$3+2+1\ne 11$$ $\endgroup$ Commented May 25, 2013 at 5:46
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    $\begingroup$ Linear term? Surely, next-to-leading term. $\endgroup$ Commented May 25, 2013 at 5:58
  • $\begingroup$ @Ethan and @ Gerry. It will be next to the leading term. Thanks..... $\endgroup$ Commented May 25, 2013 at 6:03
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Here is the way I would show it. $A = VDV^{-1}$ which means $tr(A) = tr(VDV^{-1}).$ But trace has a cyclic property. so $tr(VDV^{-1}) = tr(DV^{-1}V) = tr(D)$ so I think your assertion is correct!

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Let $A \in M_n(k)$ and let $\overline{k}$ be an algebraic closure of $k$. Because the charateristic polynomial $p_A$ of $A$ splits into linear factors in $\overline{k}[X]$, $A$ is trigonalizable in $M_n(\overline{k})$: there exists $P \in GL_n(\overline{k})$ such that $A=PTP^{-1}$ for some triangular matrix $T \in M_n(\overline{k})$. Now $\text{tr}(A)=\text{tr}(T)$ is the sum of the eigenvalues of $T$, and a fortiori of $A$.

Notice that $\text{tr}(A)$ is a symmetric polynomial with respect to the eigenvalues of $A$ so you can express $\text{tr}(A)$ with respect to the elementary symmetric polynomials and finally with respect to the coefficients of $p_A$. Because $p_A \in k[X]$, $\text{tr}(A) \in k$.

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Since all eigenvalues exist in the base field, $A$ will be similar to a matrix $J$ in Jordan Canonical Form, with eigenvalues of $A$ along the diagonal, and assertion easily follows. (Hint: Use $tr(AB) = tr(BA)$ for $A, B$ square matrices)

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