Multiplication of incompatible matrices

[attaran]

Dear members,

I have a rather silly question.
As we all know only the compatible matrices can be multiplied. My derivation of some Finite Element formulation has, however, led me to the multiplication of two incompatible matrices.

I was wondering if we could make these incompatible matrices, compatible by some factorization techniques. Has anyone ever encountered such a problem?

I appreciate if someone could help me in this respect.

Regards,

-- Hamid Attaran

 

[chiro]

Hey attaran and welcome to the forums.

Can you provide more information? What dimensions are the matrices? Are the matrices special types? (Like diagonal, triangular, singular, non-singular, have specific determinant, eigenvalues, eigenvectors and so on)

 

[attaran]

Thanks for your reply, Chiro.

Some are diagonal and some are not.
For one case I have reached from:
[A]_8x2._2x2.[C]_2x4.[D]_4x1.[E]_2x8
to:
[]_8x1.[]_2x8

which obviously are not compatible for matrix multiplication.

These are populated as follows:

A = [A₁₁ 0
0 A₂₂
...
0 A₈₂]

B = [B₁₁ 0
0 B₂₂ ]

C = [C₁₁ C₁₂ C₁₃ C₁₄
C₂₁ C₂₂ C₂₃ C₂₄]

D = [D₁₁
D₂₁
D₃₁
D₄₁]

E= [E₁₁ 0 E₁₃ 0 ... E₁₇ 0
E₂₁ 0 E₂₃ 0 ... E₂₇ 0]

I hope I have provided enough information.

regards,
-- Hamid

 

[DonAntonio]

Thanks for your reply, Chiro.

Some are diagonal and some are not.
For one case I have reached from:
[A]_8x2._2x2.[C]_2x4.[D]_4x1.[E]_2x8
to:
[]_8x1.[]_2x8

which obviously are not compatible for matrix multiplication.

These are populated as follows:

A = [A₁₁ 0
0 A₂₂
...
0 A₈₂]

B = [B₁₁ 0
0 B₂₂ ]

C = [C₁₁ C₁₂ C₁₃ C₁₄
C₂₁ C₂₂ C₂₃ C₂₄]

D = [D₁₁
D₂₁
D₃₁
D₄₁]

E= [E₁₁ 0 E₁₃ 0 ... E₁₇ 0
E₂₁ 0 E₂₃ 0 ... E₂₇ 0]

I hope I have provided enough information.

regards,
-- Hamid

Well, either (1) you define a new multiplication between non-compatible matrices, or (2) you embed all your matrices into

one single set of square matrices, perhaps by adding zeros to their rows or columns, or (3) you plainly cannot multiply those matrices among them.

I wonder why you think, or feel, that you must multiply those matrices...?


DonAntonio

 

[HallsofIvy]

The point is that if your method arrives at the multiplication of two "incompatible" matrices, then this cannot be ordinary multiplication of matrices. How you would alter the usual definition of multiplication of matrices would depend upon what you want this multiplication to mean. And that can only be determined by the precise problem you are dealing with.

 

[AlephZero]

Most likely you have got confused about something.

Computer programs for the finite element method often store data in 2-d arrays which are NOT "matrices" in the mathematical sense. For example the 6 independent compoents of a symmetric 3x3 tensor (for eaxmple stress and strain) are often stored in a 6x1 vector. Or 21 independent constants for an arbitrary anisotropic material might be stored in a symmetric 6x6 matrix, when it is really a 3x3x3x3 fourth-order tensor with a large number of symmetry relations between the 81 terms.

 

[attaran]

DonAntonio, HallsofIvy and AlephZero, Thanks for your comments and help.

As I was discussing my problem with a mathematician, he advised me to use the "Tensor Product" instead and after a quick survey, looks like that I can use a special case of tensor product, called "Kronecker product". According to Wikepedia(en.wikipedia.org/wiki/Kronecker_product)

"If A is an m-by-n matrix and B is a p-by-q matrix, then the Kronecker product A ⊗ B is the mp-by-nq block matrix ".

Thanks and regards,

-- Hamid