Explain why iterative methods are necessary to find eigenvalues and eigenvectors

Question

Explain why iterative methods are necessary to find eigenvalues and eigenvectors of matrices. Write a function in MATLAB that takes as input a square matrix A, a guess for an eigenvalue rho and returns as output the eigenvalue and eigenvector found using the shift-and-invert strategy. Use basic programming, except you may use MATLAB's built in matrix computation commands such as addition, subtraction, "division, " and multiplication, if needed. To test your program, if you enter the matrix A = [1, 2, 3: 4, 5, 6: 7, 8, 10] and rho = 1, you should find the eigenvalue 0.19825 and the eigenvector (-0.33455, 1, -0.57726).

Explanation / Answer

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4) Any monic polynomial is the characteristic polynomial of its companion matrix. Therefore, a general algorithm for finding eigenvalues could also be used to find the roots of polynomials.

The Abel-Ruffini theorem shows that any such algorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional powers.

For this reason algorithms that exactly calculate eigenvalues in a finite number of steps only exist for a few special classes of matrices. For general matrices, algorithms are iterative, producing better approximate solutions with each iteration.

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