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Matrix operations, array operations

The following matrix operations are available in MATLAB:

 +    addition  
 -    subtraction  
 *    multiplication 
 ^    power 
 '    transpose 
 \    left division 
 /    right division
These matrix operations apply, of course, to scalars (1-by-1 matrices) as well. If the sizes of the matrices are incompatible for the matrix operation, an error message will result, except in the case of scalar-matrix operations (for addition, subtraction, and division as well as for multiplication) in which case each entry of the matrix is operated on by the scalar.

The ``matrix division'' operations deserve special comment. If A is an invertible square matrix and b is a compatible column, resp. row, vector, then

 x=A \ b
is the solution of A*x=b and, resp.,
is the solution of x*A=b . In left division, if A is square, then it is factored using Gaussian elimination and these factors are used to solve A*x=b. If A is not square, it is factored using Householder orthogonalization with column pivoting and the factors are used to solve the under- or over- determined system in the least squares sense. Right division is defined in terms of left division by

    b/A=(A' \ b')'
Array operations. The matrix operations of addition and subtraction already operate entry-wise but the other matrix operations given above do not-they are matrix operations. It is important to observe that these other operations,

  *  ,   ^  ,  \  , and /,
can be made to operate entry-wise by preceding them by a period. For example, either


will yield [1,4,9,16]. Try it. This is particularly useful when using Matlab graphics.

Wed Mar 13 19:15:55 MET 1996