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2007 Feb 20 09:57

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Re: [vox] OT: matrix operations
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Re: [vox] OT: matrix operations



On 02/20/2007, at 7:57, Henry House wrote:

Why does one never hear about a scalar addition operation? It would make sense to me to define the following:

[1 2] [3 4]
[3 4] + 2 = [5 6]
Well, you could define such an operation, but mathematically it isn't necessary; the above operation is equivalent to:

[1 2]    [1 1]   [3 4]
[3 4] + 2[1 1] = [5 6]
Defining an extra addition operation doesn't have any place in the axioms governing vector spaces, either, so it would just be 'something extra'. Assuming it's defined as my above example illustrates, I can't think of any way to break 'scalar addition' without violating the underlaying ring axioms, so it should be solid mathematically.

Similarly, why is there no scalar exponentiation on vectors?
Do you mean something like the following:

[1 2] [1 2][1 2][1 2]
( [3 4] )^3 = [3 4][3 4][3 4]

If that's the case, this sort of operation has been around for quite some times, and stems from the set of NxN matrices being a ring. If you're talking about non-square matrices, than exponentiation would be undefined, because the product:

[1 2 3][1 2 3]
[4 5 6][4 5 6]

is undefined.

Hope that helps.
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