Addition of 2 matrices with different dimensions

Question

Christos Tsallis 2021-3-12

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评论： Christos Tsallis 2021-3-12

Hello,

If we want to add 2 matrices in maths, their dimensions must be the same.

I was wondering how MATLAB can add x=[1;5;9] which has dimensions of 3x1 with y=[9 2 8] which has dimensions 1x3 and the result is x+y=[10 3 9;14 7 13;18 11 17] which has dimensions 3x3.

Thanks for your time, appreciate your help.

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Answer 1

Allen 2021-3-12

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In this instance your are describing, MATLAB is assuming element-wise addition between the rows and columns. Where

Where, x =

[x1

x2

x3];

and y = [y1 y2 y3];

The result of x+y then becomes:

[x1+y1 x1+y2 x1+y3

x2+y1 x2+y2 x2+y3

x3+y1 x3+y2 x3+y3]

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Star Strider 2021-3-12

This is due to ‘Automatic Implicit Expansion’ introduced in R2016b.

See ‘Implicit Expansion’ under ‘Mathematics’ ina the Release Notes.

Christos Tsallis 2021-3-12

Thanks a lot, but is this mathematically correct?

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Answer 2

Steven Lord 2021-3-12

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If we want to add 2 matrices in maths, their dimensions must be the same.

I believe most texts accept a slight generalization of that, to allow adding scalars to matrices that are not 1-by-1. Scalar expansion has been part of MATLAB for longer than I've been at MathWorks (nearly 20 years.) It's probably in Cleve's original Fortran MATLAB.

A = magic(3)
A = 3×3
     8     1     6
     3     5     7
     4     9     2
B = A + 1 % If we applied the idea you stated this should be A + ones(3)
B = 3×3
     9     2     7
     4     6     8
     5    10     3

Implicit expansion is a generalization of that behavior. It avoids having to repmat the vectors to a common size we can compute, thus saving memory. After all, if A in my example above took up 1 GB of space (which would mean B would also take 1 GB of space) do you really want to have to allocate a temporary 1 GB matrix all of whose elements are 1?

Thanks a lot, but is this mathematically correct?

I won't tell the Mathematics Police if you don't.