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Summary

Chapter 11: Many facts about matrices can be revealed (or questions about them answered) by performing a suitable transformation that puts them into a special form. Such a form typically contains many zero entries in strategic locations. In this chapter, we show that every square complex matrix is unitarily similar to an upper triangular matrix. This is a powerful result with a host of important consequences.

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

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11 - Unitary Triangularization and Block Diagonalization

Summary

A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving

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11 - Unitary Triangularization and Block Diagonalization

Summary

A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving