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HasuChObe
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Why does performing unitary similarity transforms on a matrix eventually cause it to converge to Schur form?
QR iteration is a numerical method used to compute the eigenvalues of a matrix. It is an efficient and accurate algorithm that iteratively transforms a matrix into a similar matrix with the same eigenvalues, but in a more desirable form for computing.
QR iteration works by decomposing a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). These matrices are then multiplied together, and the resulting matrix is used as the input for the next iteration. This process is repeated until the matrix converges to a diagonal matrix, which contains the eigenvalues of the original matrix.
The main advantage of QR iteration is its efficiency and accuracy in computing eigenvalues. It does not require any assumptions about the matrix (such as being symmetric or diagonalizable), and it can handle matrices of any size. Additionally, QR iteration is a stable algorithm, meaning small changes in the input matrix will not greatly affect the output.
QR iteration is most commonly used when computing the eigenvalues of large, non-symmetric matrices. It is also useful when the eigenvalues of a matrix are needed to solve other numerical problems, such as linear systems or differential equations.
QR iteration may not be the most efficient method for computing eigenvalues of smaller, symmetric matrices. In these cases, other methods such as power iteration or Jacobi iteration may be more suitable. Additionally, QR iteration may not converge for matrices with repeated eigenvalues or for matrices with eigenvalues that are close in value.