Matrix inverse
 

MInverse function can return answer if array has an equal number of rows and columns. Is there a known trick (any function or VBA) to invert matrix which does not has an equal number of rows and columns?

Matrix inverse
 

Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.

Alan Beban

Chee Beng wrote:
MInverse function can return answer if array has an equal number of rows and columns. Is there a known trick (any function or VBA) to invert matrix which does not has an equal number of rows and columns?

Matrix inverse
 

"Alan Beban" wrote...
Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.

If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

http://mathworld.wolfram.com/Moore-P...ixInverse.html

Chee Beng wrote:
MInverse function can return answer if array has an equal number
of rows and columns. Is there a known trick (any function or VBA)
to invert matrix which does not has an equal number of rows and
columns?

Do you mean a Moore-Penrose pseudoinverse? If so, follow the link above, and
see formula (7). Otherwise, follow this link.

http://mathworld.wolfram.com/Pseudoinverse.html

It's possibe Octave (the GNU Project's rough work-alike for MatLab) includes
source code for generalized matrix inverses.

Matrix inverse
 

Harlan Grove wrote:
"Alan Beban" wrote...

Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.


If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

http://mathworld.wolfram.com/Moore-P...ixInverse.html

Enlighten us. Can you give us an example of a non-square matrix that has
an inverse?

Alan Beban

Matrix inverse
 

Harlan Grove wrote:
"Alan Beban" wrote...

Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.


If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

http://mathworld.wolfram.com/Moore-P...ixInverse.html

You outdo yourself. It's amazing, and amusing, the lengths to which
you'll go to take exception to anything I post.

A matrix inverse is defined such that if B is the inverse of A, then
AB = BA = I, the identity matrix or unit matrix.

A matrix has an inverse if and only if it is nonsingular.

A nonsingular matrix is always a square matrix whose rank is equal to
its order and whose determinant, therefore, is not zero.

Now, which part of that suggests that I don't know what I'm talking
about without a degree in some particular field, and which is
contradicted by any material at the link you provided?

Alan Beban

Matrix inverse
 

"Alan Beban" wrote...
Harlan Grove wrote:
"Alan Beban" wrote...
Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.

If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

http://mathworld.wolfram.com/Moore-P...ixInverse.html

You outdo yourself. It's amazing, and amusing, the lengths to which
you'll go to take exception to anything I post.
...

You could have asked the OP what the OP intended. Generalized matrix inversion
is an infrequently used operation in certain forms of econometric modeling and
general regression. There's an outside chance that's what the OP meant.

A matrix inverse is defined such that if B is the inverse of A, then
AB = BA = I, the identity matrix or unit matrix.
...
Now, which part of that suggests that I don't know what I'm talking
about without a degree in some particular field, and which is
contradicted by any material at the link you provided?

Strictly speaking, only square matrices have inverses. However, it's often the
case that OPs don't use strict terminology. In this particular instance, the
OP's name and grammar suggest that English may not be his/her native language,
in which case s/he may not know the precise term 'pseudoinverse'.

This is all idle speculation.

--
To top-post is human, to bottom-post and snip is sublime.

Matrix inverse
 

Alan Beban wrote:

Harlan Grove wrote:
....
http://mathworld.wolfram.com/Moore-P...ixInverse.html
....


A matrix inverse is defined such that if B is the inverse of A, then
AB = BA = I, the identity matrix or unit matrix.

A matrix has an inverse if and only if it is nonsingular.

A nonsingular matrix is always a square matrix whose rank is equal to
its order and whose determinant, therefore, is not zero.

My abstract algebra was more years back than I care to remember, but
algebra systems, including matrices can have right or left inverses.
For instance
-3 2
2 -1
0 0
is one of infinitely many right inverses of
1 2 1
2 3 4
My vague recollection is that in any algebra, if an element has both a
right and a left inverse then they are both the same thing and simply an
inverse, as you noted.

If B is an inverse of A, then ABA = A. More generally, if ABA = A then
B is called a generalized inverse or pseudo-inverse of A, even if A is
singular. A great deal of theory underlying general linear models in
statistics is developed using pseudo-inverses, because models may be
over-specified (almost always true of ANOVA models, unless restrictions
are added to the parameters). There are infinitely many generalized
inverses of a singular matrix. The Moore-Penrose inverse is a unique
one of those generalized inverses that has the four properties noted in
Harlan's link.

Jerry

Matrix inverse
 

I understand how an inverse is defined. How is a right inverse defined?

Alan Beban

Jerry W. Lewis wrote:
Alan Beban wrote:

Harlan Grove wrote:

...

http://mathworld.wolfram.com/Moore-P...ixInverse.html

...


A matrix inverse is defined such that if B is the inverse of A, then
AB = BA = I, the identity matrix or unit matrix.

A matrix has an inverse if and only if it is nonsingular.

A nonsingular matrix is always a square matrix whose rank is equal to
its order and whose determinant, therefore, is not zero.


My abstract algebra was more years back than I care to remember, but
algebra systems, including matrices can have right or left inverses. For
instance
-3 2
2 -1
0 0
is one of infinitely many right inverses of
1 2 1
2 3 4
My vague recollection is that in any algebra, if an element has both a
right and a left inverse then they are both the same thing and simply an
inverse, as you noted.

If B is an inverse of A, then ABA = A. More generally, if ABA = A then
B is called a generalized inverse or pseudo-inverse of A, even if A is
singular. A great deal of theory underlying general linear models in
statistics is developed using pseudo-inverses, because models may be
over-specified (almost always true of ANOVA models, unless restrictions
are added to the parameters). There are infinitely many generalized
inverses of a singular matrix. The Moore-Penrose inverse is a unique
one of those generalized inverses that has the four properties noted in
Harlan's link.

Jerry

Matrix inverse
 

"Alan Beban" wrote...
I understand how an inverse is defined. How is a right inverse defined?

You can't perform web searches? Right and left inverses have to do with
mappings.

http://mathworld.wolfram.com/Invertible.html

They're related to generalized matrix inverses as follows. Given matrix A,
representing a mapping from X to Y. Its left inverse, L, is a matrix that
satisfies

L A x = x for all x in X.

Its right inverse, R, is a matrix that satisfies

A R y = y for all y in Y.

So, L A is the identity matrix for X and A R is the identity matrix for Y.
If X and Y have the same dimensions, their identity matrices are the same,
so

L A = I = A R

Takes some work to show that L = R in that case.

Matrix inverse
 

Yo, you give those with degrees far too much credit.

If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

But to 'invert the above point of view Harlan; I wouldn't suggest those with degrees always know what they are talking about?


Regards Robert

"Harlan Grove" wrote in message ...
"Alan Beban" wrote...
Unless you mean something odd by "invert matrix", there is no such
concept. Mathematically, the inverse of a matrix does not exist for
non-square matrices.

If you don't have a degree in a particular field, there's an outside chance
you don't know what you're talking about.

http://mathworld.wolfram.com/Moore-P...ixInverse.html

Chee Beng wrote:
MInverse function can return answer if array has an equal number
of rows and columns. Is there a known trick (any function or VBA)
to invert matrix which does not has an equal number of rows and
columns?

Do you mean a Moore-Penrose pseudoinverse? If so, follow the link above, and
see formula (7). Otherwise, follow this link.

http://mathworld.wolfram.com/Pseudoinverse.html

It's possibe Octave (the GNU Project's rough work-alike for MatLab) includes
source code for generalized matrix inverses.




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