How to subs in a polynomial a matrix

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

[madhan ravi]

can you explain the logic behind your result?

[Bruno Luong]

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]

Thanks Bruno didn't realise at the first sight.

Answer 1

[Astha Singh]

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.

[madhan ravi]

+1 , Thank you Astha Singh , read about this function a long time back but could recall at the time.