Beyond the Basics - A Guide to MATLAB for Beginners and Experienced Users - page 60

Solving Linear Systems
Suppose A is a nonsingular n × n matrix and b is a column vector of length n .
Then typing x=A\b numerically computes the unique solution to A*x=b .
Type help mldivide for more information.
If either A or b is symbolic rather than numeric, then x=A\b computes
the solution to A*x=b symbolically. To calculate a symbolic solution when
bothinputs are numeric, type x = sym(A)\b .

Calculating Eigenvalues and Eigenvectors

The eigenvalues of a square matrix A are calculated with eig(A) . The com-
mand [U, R] = eig(A) calculates boththe eigenvalues and eigenvectors.
The eigenvalues are the diagonal elements of the diagonal matrix R , and the
columns of U are the eigenvectors. Here is an example illustrating the use of
eig :

>>A=[3-20;2-20;011];
>> eig (A)
ans =

1
-1
2
>> [U, R] = eig(A)
U=

0 -0.4082 -0.8165

0 -0.8165 -0.4082

1.0000 0.4082 -0.4082

R=

1 0 0

0 -1 0

0 0 2

The eigenvector in the first column of U corresponds to the eigenvalue
in the first column of R , and so on. These are numerical values for the
eigenpairs. To get symbolically calculated eigenpairs, type [U, R] =
eig(sym(A)) .