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  "body": "\n## Definition\n\nLet $n, m \\ge 1$.\nLet $A$ be a real $n \\times n$ [[matrix|0VD9ZSWWJG]] and let $B$ be a real $n \\times m$ matrix.\nLet $Q$ be a real [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \\times n$ matrix, and let $R$ be a real symmetric [[positive definite|0KX66CZMBB]] $m \\times m$ matrix.\nWrite $M^T$ for the [[transpose|09DXWYRD43]] of a matrix $M$.\nThe *discrete-time algebraic Riccati equation* is\n$$X = A^T X A - A^T X B \\left(R + B^T X B\\right)^{-1} B^T X A + Q$$\nfor an unknown real symmetric positive semidefinite $n \\times n$ matrix $X$.\n\n## Remarks\n\nFor such $X$ the matrix $R + B^T X B$ is positive definite, hence [[invertible|0EB9N0S33C]], so the equation makes sense.\nA solution $X$ is *stabilising* if every eigenvalue of $A - BK$ has absolute value less than $1$, where $K = (R + B^T X B)^{-1} B^T X A$.\nIf the pair $(A, B)$ is stabilisable and the pair $(Q^{1/2}, A)$ is detectable, the equation has a unique positive semidefinite solution, and it is stabilising.\nThe optimal control of the infinite-horizon linear-quadratic regulator is then $u_k = -K x_k$.\nExample: for $n = m = 1$ and $A = B = Q = R = 1$, the equation reads $X = X - X^2/(1 + X) + 1$, that is, $X^2 - X - 1 = 0$.\nIts nonnegative solution is $X = (1 + \\sqrt{5})/2$, and then $K = X/(1 + X) = 1/X$ and $A - BK = 1 - 1/X \\approx 0.38$, so it is stabilising.\nNon-example: the other root $X = (1 - \\sqrt{5})/2$ satisfies the matrix identity but is negative, so it is not an admissible solution.\nFor $B = 0$ the equation becomes the linear [[Stein equation|FM9Y4Y6D8F]] $X - A^T X A = Q$.\n",
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