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Linear algebra and geometry
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The condition number of the matrix in the infinity norm is
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Let the matrix and the vector be given. The Matlab instruction used to solve the system is
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Let be a linear system of order , where is a tridiagonal symmetric matrix with all elements of the principal diagonal equal to and the elements on the two co-diagonals equal to . The elements of are linearly spaced numbers in . Compute the eigenvalues of and, exploiting their properties, find the solution of the linear system , by solving two triangular systems (use the most computationally efficient factorization of ). The 1-norm of the vector obtained by summing the solution of the two triangular systems is, approximately:
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Let be a square matrix of size , having the elements on the main diagonal all equal to , the elements on the first upper and lower codiagonals all equal to and the elements on the 10th upper and lower codiagonals all equal to . Compute the condition number of in infinite norm. Rounded to the first decimal digit, its value is
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Which of the proposed methods is the most efficient in order to solve the linear system , being square, dense, non singular and having small size?
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The second component of the solution of the linear system with
is:
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Let be a symmetric matrix of order with all the elements on the main diagonal equal to and those on the (lower and upper) codiagonals equal to . Let be a matrix, whose th column is defined by linearly spaced numbers in , . Solve the linear systems . The 2-norm of the vector is approximately equal to:
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Compute the factorization of the matrix A=hilb(4). Denoting by the column vectors of the canonical basis of , the permutation matrix that we obtain is:
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Direct methods
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To solve a diagonal system of size , we can use the command
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