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Unit 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.