Calcolo numerico - il Tutorial Matlab

Calcolo del determinante di una matrice

B*Aans = 1 0 00 1 00 0 1This shows that B is indeed the inverse matrix of A.' ( In some applications of linear algebra knowledge of the determinant of a matrix is required.MATLAB built-in function det is designed for computing determinants.LetA = magic(3);Determinant of A is equal todet(A)ans =-360One of the classical methods for computing determinants utilizes a cofactor expansion. For moredetails, see e.g., [2], pp. 103-114. entry of the matrix AFunction ckl = cofact(A, k, l) computes the cofactor ckl of the a klfunction ckl = cofact(A,k,l)% Cofactor ckl of the a_kl entry of the matrix A.[m,n] = size(A);if m ~= nerror('Matrix must be square') 15endB = A([1:k-1,k+1:n],[1:l-1,l+1:n]);ckl = (-1)^(k+l)*det(B);Function d = mydet(A) implements the method of cofactor expansion for computingdeterminantsfunction d = mydet(A)% Determinant d of the matrix A. Function cofact must be% in MATLAB's search path.[m,n] = size(A);if m ~= nerror('Matrix must be

square')enda = A(1,:);
c = [];
for l=1:nc
    1l = cofact(A,1,l);
    c = [c;c1l];
end
d = a*c;

Let us note that function mydet uses the cofactor expansion along the row 1 of the matrix A.

Method of cofactors has a high computational complexity. Therefore it is not recommended for computations with large matrices. Its is included here for pedagogical reasons only.

To measure a computational complexity of two functions det and mydet we will use MATLAB built-in function flops. It counts the number of floating-point operations (additions, subtractions, multiplications and divisions).

Let
A = rand(25);
be a 25-by-25 matrix of uniformly distributed random numbers in the interval ( 0, 1 ).

Using function det we obtain
flops(0)
det(A)
ans =-0.1867
flops
ans = 10100

For comparison, a number of flops used by function mydet is
flops(0) 16
mydet(A)
ans =-0.1867
flops
ans = 223350

The adjoint matrix adj(A) of the matrix A is also of interest in linear algebra (see, e.g., [2],p.108).

function B = adj(A)
% Adjoint matrix B of the

square matrix A.

[m,n] = size(A);
if m ~= n
    error('Matrix must be square')
end
B = [];
for k = 1:n
    for l=1:n
        B = [B;cofact(A,k,l)];
    end
end
B = reshape(B,n,n);

The adjoint matrix and the inverse matrix satisfy the equation -1A = adj(A)/det(A) (see [2], p.110 ). Due to the high computational complexity this formula is not recommended for computing the inverse matrix.)

The 2-norm (Euclidean norm) of a vector is computed in MATLAB using function norm.

Let a = -2:2
a = [-2 -1 0 1 2]
The 2-norm of a is equal to
twon = norm(a) = 17
twon = 3.1623

With each nonzero vector one can associate a unit vector that is parallel to the given vector. For instance, for the vector a in the last example its unit vector is

unitv = a / twon
unitv = [-0.6325 -0.3162 0 0.3162 0.6325]

The angle between two vectors a and b of the same dimension is computed using the formula θ = arccos(a.b/||a|| ||b||), where a.b stands for the dot product of a and b, ||a|| is the norm of the vector a and arccos is the inverse cosine function.

Let

It is well known that any linear transformation in n.question is represented by an m-by-n matrix A, i.e., L(x) = Ax holds true for any x.

Matrices of some linear transformations including those of reflections and rotations are discussed in detail in Tutorial 4, Section 4.3.

With each matrix one can associate four subspaces called the four fundamental subspaces. The subspaces in question are called the column space, the nullspace, the row space, and the left nullspace. First two subspaces are tied closely to the linear transformations on the finite-dimensional spaces.

Throughout the sequel the symbols (L) and (L) will stand for the range and the kernel of the linear transformation L, respectively. Bases of these subspaces can be computed easily. Recall that (L) = column space of A and (L) = nullspace of A. Thus the problem of computing the bases of the range and the kernel of a linear transformation L is equivalent to the problem of finding bases of the column space and the nullspace of a.

function span(v, varargin)
    % Test whether or not vector v is in the span of a set
    % of vectors.
    
    A = [];
    n = length(varargin);
    for i=1:n
        u = varargin{i};
        u = u';
        A = [A u(:)];
    end
    
    v = v';
    v = v(:);
    
    if rank(A) == rank([A v])
        disp('Given vector is in the span.')
    else
        disp('Given vector is not in the span.')
    end
end

implements a well-known method for finding the transition matrix.
function V = transmat(T, S)
% Transition matrix V from a vector space having the ordered
% basis T to another vector space having the ordered basis S.
% Bases of the vector spaces are stored in columns of the
% matrices T and S.
[m, n] = size(T);
[p, q] = size(S);
if (m ~= p) | (n ~= q)
error('Matrices must be of the same dimension')
end
V = rref([S T]);
V = V(:,(m + 1):(m + n));

V = transmat(T, S)
V = 3 4
    1 2

[x]_T