MATLAB by rudra pratap

138 Applications
Step 2: Extract what you need: In the output list, V is an n x n matrix whose
columns are eigenvectors and D is an n x n diagonal matrix that ha."> the
eigenvalues of A on its diagonal. The function eig can also be utied with one
output argument, e.g., lams=eig(A) , in which case the function gives only
the eigenvalues in the vector lams.
After computing the eigenvalues and eigenvectors, you could check a few things
if you wish: Are the eigenvalues and eigenvectors ordered in the output? Check
by substituting in eqn. (5.2). For example, let us check the second eigenvalue and
second eigenvector and see if, indeed, they satisfy Av = .\v.
>> v2 = V(:,2 );
>> lam2 = 0(2,2) ;
>> A*v2 - lam2*v2
% extract the second column from V
% extract the second eigenvalue from D
% check the difference between A*v and lambda*v
Here, we have used two commands for clarity. You could, of course, combine
them into one: A*V (: , 2) -D (2, 2) *V (: , 2). Note that the last command may not
result into zero vector, as you expect, but it will have very small numbers.
5.1.4 Matrix factorizations
MATLAB provides built-in functions for several matrix factorizations (decompositions):
1. LU factorization: The name of the built-in function is lu. To get the LU
factorization of a square matrix A, type the command
[L ,U] = lu(A) ;
MATLAB returns a lower triangular matrix L and an upper triangular matrix
U such that L*U=A. It is also possible to get the permutation matrix as an
output (see the on-line help on lu) .
2. QR factorization: The name of the built-in function is qr. Typing the
command
[Q , R] = qr (A) ;
returns an orthogonal matrix Q and an upper triangular matrix R such that
Q*R=A. For more information, see the on-line help.
3. Cholesky factorization: If you have a positive definite matrix A, you can
factorize the matrix with the built-in function chol. The command
R = chol(A) ;
produces an upper triangular matrix R such that R' *R=A.
4. Singular value decomposition (svd): The name of the built-in function
is svd. If you type
[U ,D,V] = svd(A) ;
MATLAB returns two orthogonal matrices, U and V, and a diagonal matrix
D, with the singular values of A as the diagonal entries, such that U*D*V=A.