Characteristic polynomial: Difference between revisions

Revision as of 02:58, 12 April 2015

Revision as of 04:58, 13 April 2015

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Using the language of [[exterior algebra]], one may compactly express the characteristic polynomial of an ''n''×''n'' matrix ''A'' as

: <math> p_A (t) = \sum_{k=0}^n t^{n-k} (-1)^k \operatorname{tr}(\Lambda^k A) </math>

where tr(Λ^''k''A) is the [[Trace (linear algebra)|trace]] of the ''k''^th exterior power of ''A'', withwhich has dimension <math>\tbinom nk</math>,. This andtrace may be evaluated explicitlycomputed as the sum of all [[principal minor]]s of ''A'' of size ''k''; when the [[characteristic (algebra)|characteristic]] is 0 it may alternatively be computed as a single determinant, that of the {{math|''k''×''k''}} matrix,

:<math>\frac{1}{k!}

\begin{vmatrix} \operatorname{tr}A & k-1 &0&\cdots & \\