Mathematical formula for matrix concatenating

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Nemanjavuk69

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Is there a mathematical formula for concatenating two matrices?

I have this homework where I have to concatenate two matrices, namely matrix A and matrix B, they are both augmented matrices. I know how to solve the problem via Maple as shown in the picture below


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Now I am curious, how would one do it in hand?
 
Subhotosh Khan said:
In simple terms, concatenation means joining.

Look at your input matrices (to be concatenated) and output matrix. The process should be obvious - no mathematical operation involved here.
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Well, I can see that I can simply just "link" them together, having A "on top" and B "under", however, there should be some kind of mathematical theory or formula going on here? I am in a Linear Algebra class.

Also, I am using the book "linear algebra and its applications 6 global edition by David C. Lay & Steven R. Lay & Judi J. McDonald"
 
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Nemanjavuk69 said:
Well, I can see that I can simply just "link" them together, having A "on top" and B "under", however, there should be some kind of mathematical theory or formula going on here? I am in a Linear Algebra class.

Also, I am using the book "linear algebra and its applications 6 global edition by David C. Lay & Steven R. Lay & Judi J. McDonald"
Click to expand...
Some operations are just a way of writing something down that has value. The point about concatenating isn't about how you join the matrices, it's the properties the resulting matrix has. The definition is pretty much arbitrary, just as most definitions are.

-Dan
 
Nemanjavuk69 said:
Is there a mathematical formula for concatenating two matrices?

I have this homework where I have to concatenate two matrices, namely matrix A and matrix B, they are both augmented matrices. I know how to solve the problem via Maple as shown in the picture below
View attachment 34402
Now I am curious, how would one do it in hand?
Click to expand...
I think, rather than "how to do it by hand", which would just mean how to concatenate two matrices when writing them on paper, you are asking if there is a mathematical notation (as opposed to the notation used by Maple, or others), or anything interesting about it as an operation.

This is often denoted by a matrix whose elements are matrices. (See here, for example, and here.) But I don't know that this is commonly thought of as an operation, or that there is any deep theory necessitating a special notation. For example, I don't think it's significant to think of concatenation as an associative operation.
 
How about:
[math]C = P_1 A + P_2 B[/math]where
[math]P_1 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \\ \end{array} \right) \;\;\; P_2 = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \\ \end{array} \right)[/math]
But if you were to implement it in software you would be wasting quite a few cycles unnecessarily :)
 
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