AATA Python/MATLAB recap

Section 5.5 Python/MATLAB recap

Eigenvalues and eigenvectors. MATLAB/Python has its own function to evaluate eigenvalues and eigenvectors of a matrix. Given a

$n\times n$ matrix A, one can fill up a vector e with the list of all eigenvalues with the command e=eig(A).

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By writing, instead, [V,D]=eig(A), the variables V and D are filled with two

$n\times n$ matrices. The columns of the first are all eigenvectors of length 1. The second is a diagonal matrix whose diagonal elements are the eigenvalues of A so that the

$k$ -th column of V is en eigenvector of A with eigenvalue equal to the

$k$ -th element of the diagonal of D.

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clear all
format long
A = [ 1 2; 3 4]
e = eig(A)
[V,D] = eig(A)

Norms. In this Chapter's code we evaluated explicitly the Euclidean norm of a 2-components vector v. We could have accomplish the same thing using MATLAB/Python's own function norm(v,1). In general, norm(v,p) gives the

$p$ -norm

$\begin{equation*} \left[|v_1|^p+\dots+|v_n|^p\right]^{(1/p)}. \end{equation*}$

In particular, the taxicab norm is given by norm(v,1) and the Euclidean one by norm(v,2). Below we re-write the power method code using this function:

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clear all
format long
A=[ 1 2; 3 4]
v0=[7,-2]';
v = v0;
d = 1;
tol=1e-4;
i=0;
while d > tol
    i = i+1;
    v0 = v;
    z = A*v;
    v = z/norm(z,1);
    d = norm(v-v0,1);
end
iterations = i
distance = d        
lambda = v'*A*v/(v'*v)
eigenvector = v
eig(A)

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QR decomposition. MATLAB/Python can evaluate QR decompositions. The command [Q,R]=qr(A) gives the standard QR decomposition of a square matrix

$A\text{,}$ namely

$A=QR\text{.}$ The command [Q,R,P]=qr(A) gives the 'column pivoting' QR decomposition of a square matrix

$A\text{.}$ In this case,

$AP=QR\text{,}$ where

$P$ is a permutation matrix. It makes sense to use a pivoted decomposition when

$A$ is close to being degenerate.