1zcp(1) User Commands zcp(1)
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6 zcp - characteristic and minimal polynomial of a matrix
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9 zcp [OPTIONS] <File>
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12 This program reads in a square matrix and calculates its characteristic
13 or minimal polynomial. With no options, the characteristic polynomial
14 is computed in a partially factored form (see below). With -m the
15 polynomial is split into irreducible factors. Without -G, the output is
16 in text format. Each line contains one factor of the characteristic or
17 minimal polynomial. The -G option may be used to generate output which
18 is readable by the GAP computer program. The output, then, is a
19 sequence of sequences of finite field elements, representing the coef‐
20 ficients of the factors in ascending order.
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23 -Q Quiet, no messages.
24 -V Verbose, more messages.
26 -T <MaxTime>
28 Set CPU time limit
29 -G Produce output in GAP format.
31 -m Calculate the minimal polynomial.
33 -f Factor the polynomial.
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37 The characteristic polynomial of a matrix A is computed by constructing
38 a sequence 0=U_0<U_1<...<U_n=V of A-invariant subspaces, where
39 U_i/U_(i-1) is A-cyclic. In the ith step, U_i is constructed by choos‐
40 ing a random vector uϵV\U_(i-1) and calculating u,uA,uA^2,... until
41 some linear combination of these vectors falls into U_(i-1). This
42 yields a polynomial p_i(x) with up_i(A)ϵU_(i-1). p_i(x) is the charac‐
43 teristic polynomial of A on U_i/U_(i-1), and the full characteristic
44 polynomial of A is the product of all p_i's.
45 The algorithm for the minimal polynomial uses the same technique of
47 constructing a sequence (U_i) of invariant subspaces. In the ith step,
48 images uA,uA^2,... of a seed vector u are calculated, until a linear
49 combination of these vectors vanishes (this is the main difference to
50 the algorithm above). This yields a polynomial p_i(x) of minimal
51 degree with up_i(A)=0, and the minimal polynomial of A is the least
52 common multiple of the p_i's.
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55 File A square matrix.
56MeatAxe 2.4.24 zcp(1)