# Can matrix element be multiplication of two matrices

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Suppose I have a matrix P = \begin{bmatrix}x⊗x&x⊗y\\y⊗x&y⊗y\end{bmatrix} which is equal to another matrix Q = \begin{bmatrix}a&b\\c&d\end{bmatrix}.

Is it possible to solve for x and y.

Since xy and yx are equal to b and c and are not the same, the elements in the matrix P are outer product of two matrices x and y.

Is it possible to solve for x and y such that xy and yx can be made predictable.

Kindly let me know if I am not clear.

Thank you for the kind help.

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31 minutes ago, kemfys said:

# Can matrix element be multiplication of two matrices

We normally do  this the other way round.

It is called expansion of determinants.

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3 hours ago, kemfys said:

Kindly let me know if I am not clear.

It is given that $$b$$ and $$c$$ are not equal?

From which ring(?) do you take your $$x$$ and $$y,$$ and how is $$x \otimes y$$ defined? Do $$x$$ and $$y$$ have to be $$2 \times 2$$ matrices?

Edited by taeto
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Hi taeto,

Thank you for kind response.

I am not a mathematician, so I am not sure how to respond to the question based on ring.

I am a physical chemist and I have some problem with me for which I am trying to develop a model using matrices. And my model requires that multiplication between two parameters be non-commutative and therefore I have chosen matrices. And to preserve the uniqueness in the result of matrix multiplication between tow matrices, I have chosen the multiplication to be outer product.

Therefore, it is I who defined xy = b and yx = c. And the matrices x and y are  2×2 matrices. Actually, the intention is to extend it to any dimension.

Please let me know if I am not clear.

Thank you

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Hi kemfys,

actually that is exactly what a ring means. It is possible to add elements, and the addition is commutative. It is possible to multiply elements, but the product is not necessarily commutative. Matrices over fields like $$\mathbb{Q},$$  $$\mathbb{R},$$ and $$\mathbb{C},$$ are examples of rings. To say that a matrix is "over a field" means that all the entries of the matrix belong to this field. In the case of a $$2\times 2$$ matrix this would be all four entries. Matrices over a commutative ring like $$\mathbb{Z}$$ still form a ring themselves.

I think that I get that you want to consider matrices over rings that are not necessarily fields, or even commutative rings like $$\mathbb{Z}.$$ Maybe you should simply work through some examples to experience how it works. It seems that if you have $$2\times 2$$ matrices $$x$$ and $$y$$ as variables and you require your matrix $$P$$ to equal the matrix $$Q,$$ then it leaves you with four equations (expressions that should be equal to $$a,b,c,d$$) in eight unknowns (the entries of the matrices $$x$$ and $$y$$). In the general case the answer should be yes, you can find a solution. Though usually the complete solution consists in a 4-dimensional solution space.

Edited by taeto

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