? elimination to identity
Cheng Cosine
GIven a regular matrix A, we can conduct a series of operations so
that
A becomes an indntity matrix. This series of operations can be
expressed
in terms of a product of matrices: A*P_1*..*P_N = I, I is identity.
Suppose the dimension of A is N, is the max number of operation
matrices N as well? What is the name of the theorem ensuring this?
Can this be generalized to converting A into another matrix B? Also,
can these operation matrices be limited to be all unitary matrices?
Thanks,
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? elimination to identity Hi: GI--INTERRUPT TO
NOTE CAPITAL I--ven a regular matrix A, we can conduct a series of
operations so that A becomes an indntity matrix. This series of
operations can be expressed in terms of a product of matrices:
A*P_1*..*P_N = I, I is identity. Suppose the dimension of A is N, is
the max number of operation matrices N as well? What is the name of
the theorem ensuring this?... more » By Cheng Cosine - 1:05am - 1
new of 1 m
Cheng Cosine wrote:
> Hi:
>
> GIven a regular matrix A, we can conduct a series of operations so
> that
>
> A becomes an indntity matrix. This series of operations can be
> expressed
>
> in terms of a product of matrices: A*P_1*..*P_N = I, I is identity.The/or/e=(M)C^2. Musatov.
T
By
Coi
>ff
M. Michael Musatov
>
> - Show quoted text -
This text was not in my reply but appeared after it was published:
T
By
Coi
- Hide quoted text -
- Show quoted text -
>ff
Maarten Bergvelt
> Hi:
>
> GIven a regular matrix A, we can conduct a series of operations so
> that
>
> A becomes an indntity matrix. This series of operations can be
> expressed
>
> in terms of a product of matrices: A*P_1*..*P_N = I, I is identity.
>
> Suppose the dimension of A is N, is the max number of operation
>
> matrices N as well?
If A is 2x2 can you reduce it to the identity by just 2 column operations?
> What is the name of the theorem ensuring this?
False?
> Can this be generalized to converting A into another matrix B? Also,
What is this?
> can these operation matrices be limited to be all unitary matrices?
Can you bring a 2x2 matrix to the identity by using just unitary matrices?
--
Maarten Bergvelt
Ray Vickson
> On 2009-08-13, Cheng Cosine <asec...@gmail.com> wrote:
>
> > Hi:
>
> > GIven a regular matrix A, we can conduct a series of operations so
> > that
>
> > A becomes an indntity matrix. This series of operations can be
> > expressed
>
> > in terms of a product of matrices: A*P_1*..*P_N = I, I is identity.
>
> > Suppose the dimension of A is N, is the max number of operation
>
> > matrices N as well?
>
> If A is 2x2 can you reduce it to the identity by just 2 column operations?
It may require three. One column operation _matrix_ turns a column
into [1 0] or [0 1]; but an operation _matrix_ performs several
arithmetical operations. So, for a non-singular 2x2 matrix A we can
reduce A to I, or a permutation of I, using two matrices E_1 and E_2.
Then we may need to perform a single row-interchange (using a
permutation matrix P) to get I.
R.G. Vickson
Ray Vickson
The reduction you speak of is Gauss-Jordan elimination. Assuming that
A is nonsingular, we get E_N*...*E_1*A = Q*I, where Q is a permutation
matrix. The E_i perform row operations, which turn the columns of A
into columns of the identity matrix; at most N columns need be so
modified, so at most N E_i matrices are needed. To truly get I on the
right you need to pre-multiply by Q^(-1), which can be written as a
product of elementary row-swapping matrices, so you end up with
P_1*P_2*...*P_r*E_N*...*E_1*A = I, where r is the number of
transpositions needed to get from a permuted identity matrix to the
identity matrix itself. So, you may need more than N matrices.
R.G. Vickson
CCC
>
> - Show quoted text -
Hmm, therefore, in terms of solving the equations, not concerning to
build an inverse,
we will need exactly N "operation" matrices for an N-dimensional
regular matrix A?
Is it for any algorithm? For example, there are many different
iterative linear solvers,
e.g., Gauss-Seidal or SOR. Do they all share this at most N
operations?
Thank you,
Musatov
"M. Michael Musatov" <marty....@gmail.com> wrote in message
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On Aug 12, 10:40 pm, "MeAmI.org" <Me...@vzw.blackberry.net> wrote:
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T
By
Coi
>ff
Continue forgering me, fuckhead. I find it quite amusing as I
sit in my recliner sipping my sherry and listening to the smoove
sounds of contemporary adult music star Kenny G.
Ray Vickson
No, you could need more: up to N "echelon" matrices, and then maybe a
few permutation matrices. Of course, you could always cheat and
multiply a few of those matrices together until you were left with
just N. You could also need fewer than N, since (for example) the
identity matrix itself requires 0 operations , and a matrix that
differs from the identity in just one column requires one echelon
matrix.
>
> Is it for any algorithm? For example, there are many different
> iterative linear solvers,
>
> e.g., Gauss-Seidal or SOR. Do they all share this at most N
> operations?
No. Gauss-Seidel and SOR are iterative methods that usually could
require infinitely many steps to arrive at an *exact* solution. Of
course, in practice they may work quickly to achieve full machine
precision.
R.G. Vickson
>
> Thank you,
CCC
Do we have a way to estimate the max number of iterations before an
iterative solver like SOR to be converge?
Is it at least theoretically possible to find the special case in
which
SOR will converge in less than N steps for a regular matrix of
dimension N?
Thanks,