Page 27 - MATLAB Recipes for Earth Sciences

18 2 Introduction to MATLAB

MATLAB provides standard arithmetic operators for addition, +, and
subtraction, -. The asterisk, *, denotes matrix multiplication involving in-
ner products between rows and columns. As an example, we multiply the
matrix A with a new matrix B.

B = [4 2 6 5; 7 8 5 6; 2 1 -8 -9; 3 1 2 3];
The matrix multiplication then is

C = A * B
which generates the output

C =
63 46 22 28
61 43 81 78
46 34 7 11
66 61 38 33
In linear algebra, matrices are used to keep track of the coefficients of linear

transformations. The multiplication of two matrices represents the combina-
tion of two linear transformations to one single transformation. Matrix mul-
tiplication is not communative, i.e., A*B and B*A yield different results in
most cases. Accordingly, MATLAB provides matrix divisions, right, /, and
left, \, representing different transformations. Finally, the software allows
power of matrices, ^, and complex conjugate transpose, ', i.e, turning rows
into columns and columns into rows.
In earth sciences, however, matrices are often simply used as two-di-
mensional arrays of numerical data instead of an array representing a linear
transformation. Arithmetic operations on such arrays are done element-by-
element. Whereas this does not make any difference in addition and subtrac-
tion, the multiplicative operations are different. MATLAB uses a dot as part
of the notation for these operations.
For instance, multiplying A and B element-by-element is performed
by typing
C = A .* B

which generates the output

C =
8 8 18 35
63 24 -5 12
2 3 -24 -45
18 6 6 -6