Simplify symbolic matrix determinant?

Questions
,
1

I am trying to calculate the determinant of the following matrix:

matrix.png
matrix.png2524×1741 89.3 KB

the determinant is obviously a linear expression in u, v and w. However, the term I get is

det.png
det.png4589×1750 253 KB

which does not look linear in u,v,w. I tried to use the commands “Simplify”, “Fullsimplify” to reduce numerator and denominator, but with no success. The term I get is much more complicated than the one that should be derived.
Any suggestions are welcome.
Thanks.

2

Unless you know that some or all of your terms have finite precision, you might first try replacing all 0. with 0 and see how much smaller the result is. That should eliminate things like 0.*z^7+0.*z^8, etc.

You might then try replacing all your decimal coefficients with exact rational values. That will sometimes result in much larger expressions while it eliminates some terms that are just floating point noise. Then trying Simplify[N[…]] or Expand[N[…]] on that result might collect together some terms and get rid of large rational coefficients and possibly try to expose a linear portion of your result.

3

Thank you very much Bill. When I use for example 4.5 it gives me ugly expressions, but when I use 9/2 it works great. Quite strange…

4

Hi Nir

Your expressions ei which are elements inside the Matrix are linear.
The determinant on the other hand consists of products, etc. of different ei which are not linear.

Robert

5

Thank you Robert. The determinant is linear in the elements u,v and w since they appear only in the left column. When I used rational coefficients in the entries of the matrix it works great. When I use numbers like 3.6 it gives me very long expressions.

6

Hi Nir
you could try to replace all the real coefficients of, u,v,w with symbolic constants

a[1]    0 0 0 0 0 0 0 0 0 0 0
w*a[2]  0 0 0 0 0 0 0 0 0 0 0
v*a[3]  0 0 0 0 0 0 0 0 0 0 0
z*a[4]  0 0 0 0 0 0 0 0 0 0 0
w*a[5]  0 0 0 0 0 0 0 0 0 0 0
v*a[6]  0 0 0 0 0 0 0 0 0 0 0
a[7]    0 0 0 0 0 0 0 0 0 0 0
u*a[8]  0 0 0 0 0 0 0 0 0 0 0
v*a[9]  0 0 0 0 0 0 0 0 0 0 0
w*a[10] 0 0 0 0 0 0 0 0 0 0 0
a[11]   0 0 0 0 0 0 0 0 0 0 0
u*a[12] 0 0 0 0 0 0 0 0 0 0 0

Robert

7

Use Chop[] This sets all values <10.^-10 to 0 This way all things still remaining like the 0.*z^7+0.*z^8 will go away

In[1]:= a = 10.^-21
v = a*z^7 + 0.*z^8
Chop[v]

Out[1]= 1.*10^-21

Out[2]= 0. + 1.*10^-21 z^7

Out[3]= 0
8

0 0 0 (4z)/3 5 (4z)/3 0 0 5 (4z)/3 0 0
wa[2] 9 0 0 0 0 0 0 0 5/2 0 0
va[3] 0 9 0 0 5/2 0 0 0 0 0 0
uza[4] 4z 4z 9z 0 0 0 0 0 0 0 0
wa[5] 0 0 0 9 0 2z 0 0 5/2 0 0
va[6] 0 0 0 4 9 5 4z 0 0 0 0
0 5 0 (4z)/3 0 (4z)/3 9z 0 0 (4z)/3 0 5z
ua[8] 0 0 5/2 0 0 2z 9z 0 0 0 0
va[9] 0 0 0 0 5/2 0 0 9 0 2z 0
wa[10] 0 0 0 0 0 0 0 4 9 5 4z
0 0 5 (4z)/3 0 (4z)/3 0 5z 0 (4z)/3 9z 0
ua[12] 0 0 5/2 0 0 0 0 0 0 2z 9z

If you calculate the determinant and FullSimplify you get:

27/4 z^3 (-375 + 5654 z) (v (63125 + 12 z (-15875 + 9512 z)) a[3] + 
   u (-28125 + 4 (31625 - 53286 z) z) a[4] + 
   v (169025 - 198744 z) z a[6] + 
   v (-63125 - 381390 z + 49504 z^2) a[9] + 
   w ((63125 + 12 z (-15875 + 9512 z)) a[
        2] + (-63125 - 381390 z + 49504 z^2) a[
        5] + (169025 - 198744 z) z a[10]) + 
   u (50625 + 8 z (-26175 + 13468 z)) (a[8] + a[12]))

Substitute the a with your values and it is ok, linear in u,v,w.

Robert