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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ real [[matrix|0VD9ZSWWJG]].\nThe (unshifted) *QR algorithm* builds a sequence of matrices $A_0, A_1, A_2, \\ldots$ as follows.\n\n1. Set $A_0 = A$.\n2. For $k \\ge 0$, compute a [[QR decomposition|0ERWG4NXQ9]] $A_k = Q_k R_k$, with $Q_k$ an [[orthogonal matrix|0W02QESPJQ]] and $R_k$ [[upper triangular|0N4XY7Z3NN]].\n3. Set $A_{k+1} = R_k Q_k$.\n\nThe same definition applies to complex matrices with unitary $Q_k$.\n\n## Remarks\n\nSince $R_k = Q_k^T A_k$, one has $A_{k+1} = Q_k^T A_k Q_k$, so every $A_k$ is [[similar|0E89HQATM8]] to $A$ and has the same [[eigenvalues|0BQAA2BK3Q]].\nSuppose the eigenvalues of $A$ have pairwise distinct absolute values, and a mild condition on the eigenvectors holds.\nThen the entries of $A_k$ below the diagonal tend to $0$, and the diagonal entries tend to the eigenvalues.\nExample: for $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$, $Q_0 = \\tfrac{1}{\\sqrt5}\\begin{pmatrix} 2 \u0026 -1 \\\\ 1 \u0026 2 \\end{pmatrix}$ and $R_0 = \\tfrac{1}{\\sqrt5}\\begin{pmatrix} 5 \u0026 4 \\\\ 0 \u0026 3 \\end{pmatrix}$.\nSo $A_1 = R_0 Q_0 = \\tfrac15 \\begin{pmatrix} 14 \u0026 3 \\\\ 3 \u0026 6 \\end{pmatrix}$; the trace stays $4$ and the off-diagonal entry drops from $1$ to $0.6$.\nNon-example of convergence: for the rotation $A = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$, which is orthogonal, one may take $Q_0 = A$, $R_0 = I$, so $A_1 = A$ and the sequence is constant; its eigenvalues $\\pm i$ have equal absolute value.\nPractical versions first reduce $A$ to [[Hessenberg form|YCEEWJTX56]], use shifts $A_k - \\sigma_k I = Q_k R_k$, $A_{k+1} = R_k Q_k + \\sigma_k I$, and split off converged eigenvalues.\nThe algorithm was found independently by John G. F. Francis and Vera Kublanovskaya around 1960.\n",
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