The matrix A=( -1 2 1 -2 3 1 -2 2 2 ) has a double eigen

step 1 For this exercise, we have that for the determinant of A minus lambda times I, that's equivalent

The matrix A=( -1 2 1 -2 3 1 ...

The matrix $A=\left(\begin{array}{lll}-1 & 2 & 1 \\ -2 & 3 & 1 \\ -2 & 2 & 2\end{array}\right)$ has a double eigenvalue of 1 , and so our proof of convergence of the $Q R$ algorithm doesn't apply. Does the $Q R$ algorithm find its eigenvalues?

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Key Concepts

Multiple or Repeated Eigenvalues

Matrices with repeated eigenvalues, such as those with a double eigenvalue, present challenges for convergence proofs of the QR Algorithm. Repeated eigenvalues can cause the QR iteration to converge more slowly or even disrupt the expected convergence behavior, because the usual assumptions that allow one to decompose and isolate individual eigenvalue contributions may no longer hold, affecting the effectiveness of shift strategies.

Convergence of Iterative Methods

The convergence behavior of iterative algorithms like the QR Algorithm is highly dependent on the spectral properties of the matrix. Proofs of convergence typically assume that the matrix has distinct eigenvalues, which ensure that the shifts applied in the iteration process decouple the eigenvector components, leading to a clear separation of eigenvalue estimates as the algorithm iterates.

QR Algorithm

The QR Algorithm is an iterative method used for computing the eigenvalues (and sometimes eigenvectors) of a matrix. It repeatedly decomposes a matrix into an orthogonal matrix Q and an upper triangular matrix R, then forms a new matrix from the product RQ. Over iterations, under certain conditions, the resulting sequence of matrices converges to an upper triangular matrix whose diagonal entries are the eigenvalues of the original matrix.

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