Search   Memberlist   Usergroups
 Page 1 of 1 [2 Posts]
Author Message
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Mon Jun 12, 2006 11:53 pm    Post subject: Re: Elements in Inverse Matrix of A, where a(i.j) = i^j

<tomcees_math@yahoo.com> wrote:

 Quote: Hello: I need to invert the general nxn matrix: 0^0 0^1 0^2 ... 0^(n-1) 1^0 1^1 1^2 ... 1^(n-1) 2^0 ... ... ... 2^(n-1) ... ... (n-1)^0 .... .... (n-1)^(n-1) That is, each element in the matrix is i^j where i is the row and j is the column. An example when n = 5: 1 0 0 0 0 1 1 1 1 1 1 2 4 8 16 1 3 9 27 81 1 4 16 64 256 [Note that the nature of this problem does require that we accept 0^0 as 1.] The structure of this matrix does appear to 'ask for' row reduction to find the inverse. I can do specific cases, but am looking for a general, symbolic formula for the values of the entries in the inverse given n. Can anyone provide the values of the elements in the inverse in terms of i and j, given n?

Look up "Vandermonde matrix". The determinant has a simple form, but I
don't know of a simple form for the inverse. Instead of considering
the special matrix you have, try

1 x1 x1^2 ... x1^(n-1)
1 x2 x2^2 ... x2^(n-1)
...
1 xn xn^2 ... xn^(n-1)

Sometimes generalizing leads to easier solutions.

(Mathworld says the VM can be inverted in O(n^2) operations. So
there's something there. Playing with it in Mathematica, and factoring
the results, there are some patterns.)
--Ron Bruck

Posted Via Usenet.com Premium Usenet Newsgroup Services
----------------------------------------------------------
** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **
----------------------------------------------------------
http://www.usenet.com
tomcees_math@yahoo.com
science forum beginner

Joined: 07 Jun 2006
Posts: 2

Posted: Mon Jun 12, 2006 6:04 pm    Post subject: Elements in Inverse Matrix of A, where a(i.j) = i^j

Hello:

I need to invert the general nxn matrix:

0^0 0^1 0^2 ... 0^(n-1)
1^0 1^1 1^2 ... 1^(n-1)
2^0 ... ... ... 2^(n-1)
....
....
(n-1)^0 .... .... (n-1)^(n-1)

That is, each element in the matrix is i^j where i is the row and j is
the column.

An example when n = 5:

1 0 0 0 0
1 1 1 1 1
1 2 4 8 16
1 3 9 27 81
1 4 16 64 256

[Note that the nature of this problem does require that we accept 0^0
as 1.]

The structure of this matrix does appear to 'ask for' row reduction to

find the inverse. I can do specific cases, but am looking for a
general, symbolic formula for the values of the entries in the inverse
given n.

Can anyone provide the values of the elements in the inverse in terms
of i and j, given n?

TomCee

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 1 of 1 [2 Posts]
 The time now is Wed Apr 24, 2019 2:23 pm | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Diagonalizable matrix aline Math 0 Wed Nov 29, 2006 3:08 am sign of the determinant of an augmented matrix? Mark Math 4 Thu Jul 20, 2006 1:30 am spectrum of a symmetric tridiagonal random matrix pf.buonsante@gmail.com Math 0 Wed Jul 19, 2006 9:45 am distance matrix consolidation bird Math 6 Sat Jul 15, 2006 9:05 pm Rank of a matrix with bounded elements eugene Math 3 Sat Jul 15, 2006 7:46 am