ALAFF What you will learn

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Subsection 10.1.3 What you will learn

This week, you explore practical methods for finding all eigenvalues and eigenvectors of a Hermitian matrix, building on the insights regarding the Power Method that you discovered last week.

Upon completion of this week, you should be able to

Formulate and analyze subspace iteration methods.
Expose the relationship between subspace iteration and simple QR algorithms.
Accelerate the convergence of QR algorithms by shifting the spectrum of the matrix.
Lower the cost of QR algorithms by first reducing a Hermitian matrix to tridiagonal form.
Cast all computation for computing the eigenvalues and eigenvectors of a Hermitian matrix in terms of unitary similarity transformations, yielding the Francis Implicit QR Step.
Exploit a block diagonal structure of a matrix to deflate the Hermitian eigenvalue problem into smaller subproblems.
Combine all these insights into a practical algorithm.