Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and $h_d$ is another matrix.

  • For such a set of arbitrary matrices, how can one power-series expand $\sqrt {det(g)}$ in $x$?
share|improve this question
    
use MatrixPower to evaluate g, then find its determinant using Det then take the square root of the result using Sqrt then use Series on the result. –  Nasser Jun 24 at 9:19
    
@Nasser I want a general expression for the coefficients of $x$ in $\sqrt{det (g)}$ in terms of the matrices $g_i$ and $h_d$. I want to do this for arbitrary matrices $g_i$ and $h_d$ –  user6818 Jun 24 at 9:21
1  
@user6818: I would suggest to approach the problem by first simplifying it drastically. Put g = g0 + x g1 and study the cases of 2 and 3 dimensions. Also consider the determinant itself instaed of ist Sqrt. Use Series and look at the structure of the coefficients you obtain. The results can then be cautiously generalized. MMA gives you the desired development for any concrete case. Very clumsy epressions might result. –  Dr. Wolfgang Hintze Jul 16 at 8:38
    
@Dr.WolfgangHintze I think it is safe to say that very clumsy expressions will result... –  acl Jul 16 at 10:46
add comment

1 Answer 1

For any square matrix M which is the sum of two similar matrices M = A + B the determinant can be written as a sum of determinants as follows (example for two dimensions):

det(M) = det( ( A11 + B11, A12 + B12), (A21 + B21, A22 + B22) )

= det( ( A11 + 0, A12 + 0),  (A21 + B21, A22 + B22) )
+ det( ( 0 + B11, 0 + B12),  (A21 + B21, A22 + B22) )

and, expanding the lower row similarly,

= det( ( A11 + 0, A12 + 0),  (A21 + 0, A22 + 0) )
+ det( ( A11 + 0, A12 + 0),  (0 + B21, 0 + B22) )

+ det( ( 0 + B11, 0 + B12),  (A21 + 0, A22 + 0) )
+ det( ( 0 + B11, 0 + B12),  (0 + B21, 0 + B22) )

= det( ( A11, A12),  (A21, A22) )
+ det( ( A11, A12),  (B21, B22) )
+ det( ( B11, B12),  (A21, A22) )
+ det( ( B11, B12),  (B21, B22) )

Letting B = x C gives then the expansion

det(M) = 
= det( ( A11, A12),  (A21, A22) )
+ x det( ( A11, A12),  (C21, C22) )
+ x det( ( C11, C12),  (A21, A22) )
+ x^2 det( ( C11, C12),  (C21, C22) )

Here we recognise det(A) and det(C) but also determinants of matrices mixed between A and C, more exactly, with replacement of rows.

This procedure obviously generalizes to your problem. You might wish to write it down in MMA terms.

Regards, Wolfgang

share|improve this answer
    
Dr. Wolfgang Hintze, I just read your profile. Welcome aboard, and I hope you find this "home" to your liking. By the way, you should read Markdown Editing Help -- it'll help you make your posts look spiffy. –  Mr.Wizard Jul 16 at 10:39
    
@Mr. Wizard, yes, I do like it, thanks. I read your profile as well and found you very active in the old group. Surprisingly, we had no topic in common there ... –  Dr. Wolfgang Hintze Jul 16 at 17:22
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.