1 Laws of Matrix Algebra

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1.1 6 Old Rules, 5 Still Apply

We had associative, commutative and distributive laws for scalar algebra, we can think of them as the six bullet points below. Only the multiplicative-commutative law no longer works for matrix, the other rules work for matrix as well as scalar algebra.

Associative laws work as in scalar algebra for matrix

  • \(\displaystyle (A+B)+C=A+(B+C)\)

  • \(\displaystyle (A\cdot B)\cdot C=A\cdot (B\cdot C)\)

Commutative Law works as well for addition

  • \(\displaystyle A+B=B+A\)

  • with scalars, we know \(3\cdot 4=4\cdot 3\), but commutative law for matrix multiplication does not work, Matrix \(A\cdot B\not= B\cdot A\). The matrix dimensions might not even match up for multiplication. (see below for examples)

And Distributive Law still applies to matrix

  • \(\displaystyle A\cdot (B+C)=A\cdot B+A\cdot C\)

  • \(\displaystyle (B+C)\cdot A=B\cdot A+C\cdot A\)

1.2 Example for \(A\cdot B\not= B\cdot A\)

% Non-Square
A = rand(2,3)

A = 2x3    
    0.6959    0.6385    0.0688
    0.6999    0.0336    0.3196

B = rand(3,4)

B = 3x4    
    0.5309    0.8200    0.5313    0.6110
    0.6544    0.7184    0.3251    0.7788
    0.4076    0.9686    0.1056    0.4235

% This is OK
disp(A*B)

    0.8154    1.0960    0.5847    0.9516
    0.5238    0.9076    0.4166    0.5891

% This does not work
try 
    B*A
catch ME
    disp('does not work! Dimension mismatch')
end

does not work! Dimension mismatch


% Square
A = rand(3,3)

A = 3x3    
    0.0908    0.2810    0.4574
    0.2665    0.4401    0.8754
    0.1537    0.5271    0.5181

B = rand(3,3)

B = 3x3    
    0.9436    0.2407    0.6718
    0.6377    0.6761    0.6951
    0.9577    0.2891    0.0680

% This is OK
A*B

ans = 3x3    
    0.7030    0.3441    0.2875
    1.3704    0.6147    0.5445
    0.9773    0.5431    0.5049

% This works, but result differs from A*B
B*A

ans = 3x3    
    0.2531    0.7252    0.9904
    0.3449    0.8432    1.2437
    0.1745    0.4322    0.7263

1.3 4 New Rules for Transpose

In scalar algebra, transpose does not make sense. Given matrix \(A\), \(A^T\) is the transpose matrix of \(A\) where each row of \(A\) becomes columns in \(A^T\). If \(A\) is \(M\) by \(N\), then \(A^T\) is \(N\) by \(M\).

Given matrix \(A\) and scalar value \(r\):

  • 1: \((r\cdot A)^T =r\cdot A^T\)

  • 2: \((A^T )^T =A\)

  • 3: \((A+B)^T =A^T +B^T\)

  • 4: \((A\cdot B)^T =B^T \cdot A^T\)

For the 4th rule, suppose matrix \(A\) is has \(L\) rows and \(M\) columns, and the matrix \(B\) has \(M\) rows and \(N\)columns. \((A\cdot B)\) is a \(L\) by \(N\) matrix, \((A\cdot B)^T\) is a \(N\) by \(L\) matrix. This is equal to \(B^T \cdot A^T\), where we have a \(N\) by \(M\) matrix \(B^T\) multiplied by a \(M\) by \(L\) matrix \(A^T\), and the resulting matrix is \(N\) by \(L\).

A = rand(2,3)

A = 2x3    
    0.2548    0.6678    0.3445
    0.2240    0.8444    0.7805

Atranspose = (A')

Atranspose = 3x2    
    0.2548    0.2240
    0.6678    0.8444
    0.3445    0.7805