Matrix Addition and Multiplication

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Scalar Multiplication/Division, Addition/Subtraction

If we multiply a matrix by a number, we multiply every element of that matrix by that number. Addition, subtraction, and division of a matrix with a sclar value work the same way

c = 10
c = 10
matA = rand(3,2)
matA = 3×2
    0.3111    0.1848
    0.9234    0.9049
    0.4302    0.9797
c*matA
ans = 3×2
    3.1110    1.8482
    9.2338    9.0488
    4.3021    9.7975
matA/c
ans = 3×2
    0.0311    0.0185
    0.0923    0.0905
    0.0430    0.0980
matA - c
ans = 3×2
   -9.6889   -9.8152
   -9.0766   -9.0951
   -9.5698   -9.0203
matA + c
ans = 3×2
   10.3111   10.1848
   10.9234   10.9049
   10.4302   10.9797

Addition and Subtraction

You can add/subtract together two matrixes of the same size. We can add up the two 3 by 1 vectors from above, and the two 2 by 3 matrixes from above.

colVecA = rand(3,1)
colVecA = 3×1
    0.4389
    0.1111
    0.2581
colVecB = rand(3,1)
colVecB = 3×1
    0.4087
    0.5949
    0.2622
matA = rand(3,2)
matA = 3×2
    0.6028    0.1174
    0.7112    0.2967
    0.2217    0.3188
matB = rand(3,2)
matB = 3×2
    0.4242    0.2625
    0.5079    0.8010
    0.0855    0.0292
colVecA + colVecB
ans = 3×1
    0.8476
    0.7060
    0.5203
matA - matB
ans = 3×2
    0.1787   -0.1451
    0.2034   -0.5043
    0.1362    0.2896

When using matlab, even if you add up to a single column or single row with a matrix that has multiple rows and columns, if the column count or row count matches up, matlab will broadcast rules, and addition will still be legal. In the example below, matA is 3 by 2, and colVecA is 3 by 1, matlab duplicate colVecA and add it to each column of matA (Broadcast rules are important for efficient storage and computation):

matA + colVecA
ans = 3×2
    1.0417    0.5563
    0.8223    0.4078
    0.4798    0.5768

Matrix Multiplication

When we try to multiply two matrixes together:

for example, the number of columns of matrix A and the number of rows of matrix B have to match up.

If the matrix A is has L rows and M columns, and the matrix B has M rows and Ncolumns, then the resulting matrix of

has to have L rows and N columns.

Each of the

cell in the product matrix

, is equal to:

Note that we are summing over M: row l in matrix A, and column n in matrix B both have M elements. We multiply each m of the M element from the row in A and column in B together one by one, and then sum them up to end up with the value for the lth row and nth column in matrix C.

% (3 by 4) times (4 by 2) end up with (3 by 2)
L = 3;
M = 4;
N = 2;
matA = rand(L, M)
matA = 3×4
    0.9289    0.5785    0.9631    0.2316
    0.7303    0.2373    0.5468    0.4889
    0.4886    0.4588    0.5211    0.6241
matB = rand(M, N)  
matB = 4×2
    0.6791    0.0377
    0.3955    0.8852
    0.3674    0.9133
    0.9880    0.7962
matC = matA*matB
matC = 3×2
    1.4423    1.6111
    1.2738    1.1262
    1.3214    1.3974
% (2 by 10) times (10 by 1) end up with (2 by 1)
L = 2;
M = 10;
N = 1;
matA = rand(L, M)
matA = 2×10
    0.0987    0.3354    0.1366    0.1068    0.4942    0.7150    0.8909    0.6987    0.0305    0.5000
    0.2619    0.6797    0.7212    0.6538    0.7791    0.9037    0.3342    0.1978    0.7441    0.4799
matB = rand(M, N)  
matB = 10×1
    0.9047
    0.6099
    0.6177
    0.8594
    0.8055
    0.5767
    0.1829
    0.2399
    0.8865
    0.0287
matC = matA*matB
matC = 2×1
    1.6524
    3.5895