Question: Compute the singular values of the \({\bf{4 \times 4}}\) matrix in Exercise 9 in Section 2.3, and compute the condition number \(\frac{{{\sigma _1}}}{{{\sigma _4}}}\).

Consider the matrix\(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 7}&{ - 7}\\{ - 6}&1&{11}&9\\7&{ - 5}&{10}&{19}\\{ - 1}&2&3&{ - 1}\end{array}} \right)\).

Enter the matrix\(A\)in MATLAB:

\( > > {\rm{ }}A = \left( {4,0, - 7, - 7; - 6,1,11,9;7, - 5,10,19; - 1,2,3, - 1} \right)\)

Find the singular value decomposition of\(A\).

\( > > {\rm{ }}B = svd\left( A \right);\)

\(B = \left( {\begin{array}{*{20}{c}}{27.386}\\{12.091}\\{2.6116}\\{0.001156}\end{array}} \right)\)

Find the condition number of\(A\).

\( > > C = cond\left( A \right);\)

\(C = 2.3683 \times {10^4}\)

Find the condition number\(\frac{{{\sigma _1}}}{{{\sigma _4}}}\)using singular values:

\(\begin{array}{c}\frac{{{\sigma _1}}}{{{\sigma _4}}} = \frac{{27.386}}{{0.001156}}\\ = 23,683\end{array}\)

Therefore, \(\frac{{{\sigma _1}}}{{{\sigma _4}}} = 23,683\).