Characteristic polynomial

Question

The characteristic polynomial of a square matrix \$A\$ is defined as the polynomial \$p_A(x) = \det(Ix-A)\$ where \$I\$ is the identity matrix and \$\det\$ the determinant. Note that this definition always gives us a monic polynomial such that the solution is unique.

Your task for this challenge is to compute the coefficients of the characteristic polynomial for an integer valued matrix, for this you may use built-ins but it is discouraged.

Rules

• input is an \$N\times N\$ (\$N \ge 1\$) integer matrix in any convenient format
• your program/function will output/return the coefficients in either increasing or decreasing order (please specify which)
• the coefficients are normed such that the coefficient of \$x^N\$ is 1 (see test cases)
• you don't need to handle invalid inputs

Testcases

Coefficients are given in decreasing order (ie. \$x^N, x^{N-1}, ..., x^2, x, 1\$):

[0] -> [1 0]
[1] -> [1 -1]
[1 1; 0 1] -> [1 -2 1]
[80 80; 57 71] -> [1 -151 1120] 
[1 2 0; 2 -3 5; 0 1 1] -> [1 1 -14 12]
[4 2 1 3; 4 -3 9 0; -1 1 0 3; 20 -4 5 20] -> [1 -21 -83 559 -1987]
[0 5 0 12 -3 -6; 6 3 7 16 4 2; 4 0 5 1 13 -2; 12 10 12 -2 1 -6; 16 13 12 -4 7 10; 6 17 0 3 3 -1] -> [1 -12 -484 3249 -7065 -836601 -44200]
[1 0 0 1 0 0 0; 1 1 0 0 1 0 1; 1 1 0 1 1 0 0; 1 1 0 1 1 0 0; 1 1 0 1 1 1 1; 1 1 1 0 1 1 1; 0 1 0 0 0 0 1] -> [1 -6 10 -6 3 -2 0 0]

Answer 1

SageMath, 3 bytes

5 bytes saved thanks to @Mego

fcp

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Takes a Matrix as input.

fcp stands for factorization of the characteristic polynomial,

which is shorter than the normal builtin charpoly.

Answer 2

Octave, 16 4 bytes

@BruteForce just told me that one of the functions I was using in my previous solution can actually do the whole work:

poly

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16 Bytes: This solution computes the eigenvalues of the input matrix, and then proceeds building a polynomial from the given roots.

@(x)poly(eig(x))

But of course there is also the boring

charpoly

(needs a symbolic type matrix in Octave, but works with the usual matrices in MATLAB.)

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Answer 3

Pari/GP, 8 bytes

charpoly

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---

Pari/GP, 14 bytes

m->matdet(x-m)

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Answer 4

R, 53 bytes

function(m){for(i in eigen(m)$va)T=c(0,T)-c(T,0)*i
T}

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Returns the coefficients in increasing order; i.e., a_0, a_1, a_2, ..., a_n.

Computes the polynomial by finding the eigenvalues of the matrix.

R + pracma, 16 bytes

pracma::charpoly

pracma is the "PRACtical MAth" library for R, and has quite a few handy functions.

Answer 5

Mathematica, 22 bytes

Det[MatrixExp[0#]x-#]&

-7 bytes from alephalpha
-3 bytes from Misha Lavrov

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and... of course...

Mathematica, 29 bytes

#~CharacteristicPolynomial~x&

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both answers output a polynomial

Answer 6

Haskell, 243 223 222 bytes

s=sum
(&)=zip
z=zipWith
a#b=[[s$z(*)x y|y<-foldr(z(:))([]<$b)b]|x<-a]
f a|let c=z pure[1..]a;g(u,d)k|m<-[z(+)a b|(a,b)<-a#u&[[s[d|x==y]|y<-c]|x<-c]]=(m,-s[s[b|(n,b)<-c&a,n==m]|(a,m)<-a#m&c]`div`k)=snd<$>scanl g(0<$c<$c,1)c

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Thanks to @ØrjanJohansen for helping me golf this!

Explanation

This uses the Faddeev–LeVerrier algorithm to compute the coefficients. Here's an ungolfed version with more verbose names:

-- Transpose a matrix/list
transpose b = foldr (zipWith(:)) (replicate (length b) []) b

-- Matrix-matrix multiplication
(#) :: [[Int]] -> [[Int]] -> [[Int]]
a # b = [[sum $ zipWith (*) x y | y <- transpose b]|x<-a]


-- Faddeev-LeVerrier algorithm
faddeevLeVerrier :: [[Int]] -> [Int]
faddeevLeVerrier a = snd <$> scanl go (zero,1) [1..n]
  where n = length a
        zero = replicate n (replicate n 0)
        trace m = sum [sum [b|(n,b)<-zip [1..n] a,n==m]|(m,a)<-zip [1..n] m]
        diag d = [[sum[d|x==y]|y<-[1..n]]|x<-[1..n]]
        add as bs = [[x+y | (x,y) <- zip a b] | (b,a) <- zip as bs]
        go (u,d) k = (m, -trace (a#m) `div` k)
          where m = add (diag d) (a#u)

_{Note: I took this straight from this solution}

Answer 7

Python 2 + numpy, 23 bytes

from numpy import*
poly

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Answer 8

MATL, 4 bytes

1$Yn

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This is merely a port of flawr's Octave answer, so it returns the coefficients in decreasing order, i.e., [a_n, ..., a_1, a_0]

1$Yn          # 1 input Yn is "poly"

Answer 9

CJam (48 bytes)

{[1\:A_,{1$_,,.=1b\~/A@zf{\f.*1fb}1$Aff*..+}/;]}

Online test suite

Dissection

This is quite similar to my answer to Determinant of an integer matrix. It has some tweaks because the signs are different, and because we want to keep all of the coefficients rather than just the last one.

{[              e# Start a block which will return an array
  1\            e#   Push the leading coefficient under the input
  :A            e#   Store the input matrix in A
  _,            e#   Take the length of a copy
  {             e#     for i = 0 to n-1
                e#       Stack: ... AM_{i+1} i
    1$_,,.=1b   e#       Calculate tr(AM_{i+1})
    \~/         e#       Divide by -(i+1)
    A@          e#       Push a copy of A, bring AM_{i+1} to the top
    zf{\f.*1fb} e#       Matrix multiplication
    1$          e#       Get a copy of the coefficient
    Aff*        e#       Multiply by A
    ..+         e#       Matrix addition
  }/
  ;             e#   Pop AM_{n+1} (which incidentally is 0)
]}

Answer 10

Python 3 and my dronery library, 476 bytes

λ M: C(λ f: λ t,m: t if len(t)==len(M) else f(f)(t+[n:=-sum(map(λ i: m[i][i],range(len(M))))/len(t)],matmul(M,lap(laph(__add__),m,lap(λ i: i*[0]+(n,)+(len(M)+~i)*[0],range(len(M)))))))([1],deepcopy(M))

(also using the Faddeev-LaVerrier algorithm)

Answer 11

Maple, 39 bytes

LinearAlgebra[CharacteristicPolynomial]

M := <<1, 0, 0, 1, 0, 0, 0> | <1, 1, 0, 0, 1, 0, 1> | <1, 1, 0, 1, 1, 0, 0> | <1, 1, 0, 1, 1, 0, 0> | <1, 1, 0, 1, 1, 1, 1> | <1, 1, 1, 0, 1, 1, 1> | <0, 1, 0, 0, 0, 0, 1>>;
LinearAlgebra[CharacteristicPolynomial](M, x);

Answer 12

GAP, 24 bytes

CharacteristicPolynomial

m := [[4,2,1,3],[4,-3,9,0],[-1,1,0,3],[20,-4,5,20]];;
chPolynomial := CharacteristicPolynomial(m);
coup := CoefficientsOfUnivariatePolynomial(chPolynomial);
Print(chPolynomial , "\n");
Print(coup , "\n");

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Answer 13

Maxima, 8 bytes

charpoly

A : matrix( [0, 5, 0, 12, -3, -6], [6, 3, 7, 16, 4, 2], [4, 0, 5, 1, 13, -2], [12, 10, 12, -2, 1, -6], [16, 13, 12, -4, 7, 10], [ 6, 17, 0, 3, 3, -1] );
ans:  expand (charpoly (A, lambda));
print(ans);

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