Question

For a matrix program, the Gram–Schmidt process worksbetter with orthonormal vectors. Starting with \({x_1},......,{x_p}\) asin Theorem 11, let \(A = \left\{ {{x_1},......,{x_p}} \right\}\) . Suppose \(Q\) is an\(n \times k\)matrix whose columns form an orthonormal basis for

the subspace \({W_k}\) spanned by the first \(k\) columns of A. Thenfor \(x\) in \({\mathbb{R}^n}\), \(Q{Q^T}x\) is the orthogonal projection of x onto \({W_k}\) (Theorem 10 in Section 6.3). If \({x_{k + 1}}\) is the next column of \(A\),then equation (2) in the proof of Theorem 11 becomes

\({v_{k + 1}} = {x_{k + 1}} - Q\left( {{Q^T}T {x_{k + 1}}} \right)\)

(The parentheses above reduce the number of arithmeticoperations.) Let \({u_{k + 1}} = \frac{{{v_{k + 1}}}}{{\left\| {{v_{k + 1}}} \right\|}}\). The new \(Q\) for thenext step is \(\left( {\begin{aligned}{{}{}}Q&{{u_{k + 1}}}\end{aligned}} \right)\). Use this procedure to compute the\(QR\)factorization of the matrix in Exercise 24. Write thekeystrokes or commands you use.

Short answer

The required MATLAB command is:

function ( Q R) = GramSchmidt_N(A)

(m,n) = size(A);

(U, jb) = rref(A);

x = length(jb);

B = zeros(m,x);

for i = 1:x

C(:,i)= A(:,(jb(i)));

end

B=C;

for i = 2:x

for j = 1:i-1

B(:,i) = B(:,i) - dot(C(:,i),B(:,j))/dot(B(:,j),B(:,j))* B(:,j)

end

end

for i=1:size(B,2)

TMP=B(:,i);

TMP=TMP./(sqrt(sum(TMP.^2)));

B(:,i)=TMP;

end

end

R=Q'*A

Step-by-step solution

\(QR\) factorization of a Matrix

A matrix with order \(m \times n\) can be written as the multiplication of an upper triangular matrix \(R\) and a matrix \(Q\) which is formed by applying the Gram–Schmidt orthogonalization processto the \({\rm{col}}\left( A \right)\).

The matrix \(R\) can be found by the formula \({Q^T}A = R\).

The matlab Programming

Using Gram-Schmidt orthogonalization command of MATLAB of the matrix \(A = \left( {\begin{aligned}{{}{r}}{ - 10}&{13}&7&{ - 11}\\2&1&{ - 5}&3\\{ - 6}&3&{13}&{ - 3}\\{16}&{ - 16}&{ - 2}&5\\2&1&{ - 5}&{ - 7}\end{aligned}} \right)\).

function ( Q R) = GramSchmidt_N(A)

(m,n) = size(A);

(U, jb) = rref(A);

x = length(jb);

B = zeros(m,x);

for i = 1:x

C(:,i)= A(:,(jb(i)));

end

B=C;

for i = 2:x

for j = 1:i-1

B(:,i) = B(:,i) - dot(C(:,i),B(:,j))/dot(B(:,j),B(:,j))* B(:,j)

end

end

for i=1:size(B,2)

TMP=B(:,i);

TMP=TMP./(sqrt(sum(TMP.^2)));

B(:,i)=TMP;

end

end

R=Q'*A

After using this command, the QR factorization is:

\(\left( {\begin{aligned}{{}{r}}{ - 10}&{13}&7&{ - 11}\\2&1&{ - 5}&3\\{ - 6}&3&{13}&{ - 3}\\{16}&{ - 16}&{ - 2}&5\\2&1&{ - 5}&{ - 7}\end{aligned}} \right) = \left( {\begin{aligned}{{}{r}}{\frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{{\sqrt 3 }}}&0\\{\frac{1}{{10}}}&{\frac{1}{2}}&0&{\frac{1}{{\sqrt 2 }}}\\{\frac{{ - 3}}{{10}}}&{ - \frac{1}{2}}&{\frac{1}{{\sqrt 3 }}}&0\\{\frac{4}{5}}&0&{\frac{1}{{\sqrt 3 }}}&0\\{\frac{1}{{10}}}&{\frac{1}{2}}&0&{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right)\left( {\begin{aligned}{{}{}}{20}&{ - 20}&{ - 10}&{10}\\0&6&{ - 8}&{ - 6}\\0&0&{6\sqrt 3 }&{ - 3\sqrt 3 }\\0&0&0&{5\sqrt 2 }\end{aligned}} \right)\)