1 Matrix Addition and Multiplication

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1.1 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 = 3x2    
    0.3111    0.1848
    0.9234    0.9049
    0.4302    0.9797

c*matA

ans = 3x2    
    3.1110    1.8482
    9.2338    9.0488
    4.3021    9.7975

matA/c

ans = 3x2    
    0.0311    0.0185
    0.0923    0.0905
    0.0430    0.0980

matA - c

ans = 3x2    
   -9.6889   -9.8152
   -9.0766   -9.0951
   -9.5698   -9.0203

matA + c

ans = 3x2    
   10.3111   10.1848
   10.9234   10.9049
   10.4302   10.9797

1.2 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 = 3x1    
    0.4389
    0.1111
    0.2581

colVecB = rand(3,1)

colVecB = 3x1    
    0.4087
    0.5949
    0.2622

matA = rand(3,2)

matA = 3x2    
    0.6028    0.1174
    0.7112    0.2967
    0.2217    0.3188

matB = rand(3,2)

matB = 3x2    
    0.4242    0.2625
    0.5079    0.8010
    0.0855    0.0292

colVecA + colVecB

ans = 3x1    
    0.8476
    0.7060
    0.5203

matA - matB

ans = 3x2    
    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 = 3x2    
    1.0417    0.5563
    0.8223    0.4078
    0.4798    0.5768

1.3 Matrix Multiplication

When we try to multiply two matrixes together: \(A\cdot B\) 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 \(N\)columns, then the resulting matrix of \(C=A\cdot B\) has to have \(L\) rows and \(N\) columns.

Each of the \((l,n)\) cell in the product matrix \(C=A\cdot B\), is equal to:

  • \(\displaystyle C_{l,n} =\sum_{m=1}^M A_{l,m} \cdot B_{m,n}\)

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 \(l\)th row and \(n\)th 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 = 3x4    
    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 = 4x2    
    0.6791    0.0377
    0.3955    0.8852
    0.3674    0.9133
    0.9880    0.7962

matC = matA*matB

matC = 3x2    
    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 = 2x10    
    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 = 10x1    
    0.9047
    0.6099
    0.6177
    0.8594
    0.8055
    0.5767
    0.1829
    0.2399
    0.8865
    0.0287

matC = matA*matB

matC = 2x1    
    1.6524
    3.5895