(a) Compute the singular values and condition numbers of the 2 ×2,3 ×3, and 4 ×4 Hilbert matrice

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(a) Compute the singular values and condition numbers of the...

(a) Compute the singular values and condition numbers of the $2 \times 2,3 \times 3$, and $4 \times 4$ Hilbert matrices. (b) What is the smallest Hilbert matrix with condition number larger than $10^6$ ?

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Key Concepts

Hilbert Matrix

A Hilbert matrix is a classic example of a poorly conditioned, symmetric, and positive definite matrix where each entry is given by H(i, j) = 1/(i + j - 1). Its structure makes it a common benchmark in numerical analysis to test the stability and accuracy of algorithms, especially in solving linear systems and performing matrix inversions.

Singular Values

Singular values of a matrix are the non-negative square roots of the eigenvalues of the matrix multiplied by its transpose (A?A for real matrices). They provide important quantitative measures of a matrix's behavior, especially in the context of decompositions like Singular Value Decomposition (SVD), and can indicate how the matrix scales and rotates input vectors.

Condition Number

The condition number of a matrix, often defined as the ratio of the largest singular value to the smallest singular value, measures the sensitivity of the solution of a system of linear equations to perturbations or errors in the data. A high condition number implies that the matrix is ill-conditioned, meaning that small changes in input can result in large deviations in the output, which is especially true for Hilbert matrices.

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