MaplePrimes - answers and comments on Question, Use fsolve or Matrix dissection

By PolynomialSystem

hirnyk — Sat, 07 Aug 2010 18:50:31 Z

Here is the solution of your matrix equation with the SolveTools package: MatrixEquation.mw .

Cholesky factorization

Axel Vogt — Sat, 07 Aug 2010 21:00:46 Z

You want some factorization C.Transpose(C) = A for a given matrix A.

Your matrix is symmetric, thus admits a base of Eigenvectors, which
gives a diagonal matrix delta. Taking the square root is easy and
gives the desired C, after combining all that.

Find a sheet attached.

PS: please do not post questions through pdf files - usually nobody
wants to type that in again. The best is: provide as ASCII text.

Cholesky_factorizati.mws

Cholesky factorization

acer — Sun, 08 Aug 2010 07:38:16 Z

C:=LinearAlgebra:-LUDecomposition(A,'method'='Cholesky','output'='L');

C.C^%T;

The option output=L is not necessary, as it is the default form returned for method=Cholesky. See here for lots more detail.

acer

A bit of theory

Robert Israel — Mon, 09 Aug 2010 00:49:49 Z

There are really just 6 equations, not 9, because C . Transpose(C) is symmetric. So you would expect at least 3 arbitrary variables

Note that if C is a solution, so is C U where U is any 3 x 3 orthogonal matrix (i.e. a matrix such that
U . Transpose(U) is the identity matrix). The orthogonal matrices are a 3-dimensional manifold, so that accounts for your three arbitrary variables. Moreover, the equation C . Transpose(C) = B . Transpose(B) (where these matrices are invertible) can be written as

B^(-1) . C . Transpose(B^(-1) . C) = B^(-1) . C . Transpose(C) . Transpose(B)^(-1)
= B^(-1) . B . Transpose(B) . Transpose(B)^(-1) = IdentityMatrix

so B^(-1).C is orthogonal, i.e. any two solutions (in the invertible case) are related by an orthogonal matrix.

The Cholesky decomposition (in the case of a positive definite matrix Q, which yours is) finds a lower triangular matrix L such that L . Transpose(L) = Q. Thus the general solution is C = L . U where U is an orthogonal matrix, and L can be obtained by

> L := LinearAlgebra:-LUDecomposition(Q, method=Cholesky);

                                    [2  0  0]
                                    [       ]
                               L := [2  2  0]
                                    [       ]
                                    [0  2  2]

It is`nt my C

reza_maple — Mon, 09 Aug 2010 08:40:45 Z

I read all answered and each result was correct , but it is`nt my c.

because i want to find a special c.

maybe my question wrong or .........

but i want to solve

and for this i use Q= C.C

then i want to find C,

(1)

(2)

(3)

Download MyMatrix.mw

Reza

can you described your answer

reza_maple — Sat, 07 Aug 2010 20:05:34 Z

i read your answer , it seems correct , i am new in linear algebra

can you described your answer .

thanks

Reza