How to subs in a polynomial a matrix

Question

Fernando Pérez Lara 2018-12-9

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编辑： madhan ravi 2018-12-13

Hello,

I'm trying to subs in the EQ polynomial a matrix A. The problem is when I subs this polynomial using subs(EQ, x, A)

syms x;
A = [1 2 3;
     4 5 6;
     7 8 9]
 
EQ = x^3 + 2*x^2 - 5*x + 3
subs(EQ, x, A)
OUTPUT:    1,   9,  33
          79, 153, 261
         409, 603, 849
        
WHAT I WANT:   526         641         756
              1177        1445        1713
              1828        2249        2670

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Bruno Luong 2018-12-9

编辑：Bruno Luong 2018-12-9

在 MATLAB Online 中打开

Logic? Seems pretty straigh-forward to me. Expression using power on matrix is repeated matrix multiplication (not element wise multiplication):

>> A^3
ans =
         468         576         684
        1062        1305        1548
        1656        2034        2412
>> A*A*A
ans =
         468         576         684
        1062        1305        1548
        1656        2034        2412
>> 
>> A^3 + 2*A^2 - 5*A + 3
ans =
         526         641         756
        1177        1445        1713
        1828        2249        2670

madhan ravi 2018-12-9

编辑：madhan ravi 2018-12-9

Thanks Bruno didn't realise at the first sight.

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Answer 1

Astha Singh 2018-12-13

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在 MATLAB Online 中打开

Substitution of a matrix into a polynomial using 'subs()' command, is done in element-by-element manner, which leads to the observed result.

In order to substitute a matrix in a polynomial and use the standard matrix operation rules, you would need to use 'polyvalm()' command.

I am attaching a sample code to achieve the same:

syms x;
A = [1 2 3;4 5 6;7 8 9];
EQ = x^3 + 2*x^2 - 5*x + 3;
% Get a row vector containing the numeric coefficients of the polynomial 'EQ'
b=sym2poly(EQ);
% Substitute the square matrix 'A' into the polynomial 'EQ'
polyvalm(b,A)

Please note that here the command 'polyvalm(b,A)' is equivalent to 'A^3 + 2*A^2 - 5*A + 3*eye(3)'. The constant times the identity matrix 'eye(3)' replaces the constant term of 'EQ'.

If on the other hand, the constant is simply added, as in: 'A^3 + 2*A^2 - 5*A + 3', it leads to the number 3 being added to all the elements of the matrix.