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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
\(\displaystyle A\cdot (B+C)=A\cdot B+A\cdot C\)
\(\displaystyle (B+C)\cdot A=B\cdot A+C\cdot A\)
% 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 0.7263In 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