The QR method for computing all eigenvalues

Inverse inflation and shift operations to find other eigenvalues 

it returns the largest modulus eigenvalue of (A - As the eigenvalues of (A — are (Xj — /x), this method finds the eigenvalue of smallest A, — /x - At each iteration, we must solve a linear system (A — which is done efficiently by 

LU decomposition, so that later iterations only require solving two triangular systems by substitution. 

We next consider a method to compute all eigenvalues of a matrix concurrently by transforming the matrix into a similar one whose eigenvalues are easy to calculate. The transformation is done through iterative use of QR decompositions, described below. 

Just as a matrix may be factored into ffie product of lower and upper triangular matrices, it may also be factored into the product of an orthogonal matrix Q, — Q, and an upper triangular matrix R, 

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