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160 GNU Octave18.4 Famous MatricesThe following functions return famous matrix forms.hankel (c, r)Function FileReturn the Hankel matrix constructed given the first column c, and (optionally) thelast row r. If the last element of c is not the same as the first element of r, the lastelement of c is used. If the second argument is omitted, it is assumed to be a vectorof zeros with the same size as c.A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements{ci+j−1 , i + j − 1 ≤ m;H(i, j) =r i+j−m , otherwise.hilb (n)Function FileReturn the Hilbert matrix of order n. The i, j element of a Hilbert matrix is definedas1H(i, j) =(i + j − 1)invhilb (n)Function FileReturn the inverse of a Hilbert matrix of order n. This can be computed computedexactly usingA ij = −1 i+j (i + j − 1)= p(i)p(j)(i + j − 1)( n + i − 1n − j) ( n + j − 1n − i) ( ) 2i + j − 2i − 2where( ) ( )k + n − 1 np(k) = −1 k k − 1 kThe validity of this formula can easily be checked by expanding the binomial coefficientsin both formulas as factorials. It can be derived more directly via the theoryof Cauchy matrices: see J. W. Demmel, Applied Numerical Linear Algebra, page 92.Compare this with the numerical calculation of inverse (hilb (n)), which suffersfrom the ill-conditioning of the Hilbert matrix, and the finite precision of your computer’sfloating point arithmetic.sylvester matrix (k)Return the Sylvester matrix of order n = 2 k .Function File