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  "body": "\n## Definition\n\nLet $A$ be a [[positive semidefinite|01MMVW8CKC]] $n \\times n$ complex Hermitian matrix.\nThere is exactly one positive semidefinite matrix $B$ with $B^2 = A$.\nThis matrix is the *square root* of $A$, written $A^{1/2}$ or $\\sqrt{A}$.\n\n## Remarks\n\nIf $A = U D U^*$ with $U$ unitary and $D$ diagonal with entries $d_i \\ge 0$, then $A^{1/2} = U D^{1/2} U^*$, with entries $\\sqrt{d_i}$ in $D^{1/2}$.\n$A^{1/2}$ commutes with each matrix that commutes with $A$.\nWithout the condition $B \\succeq 0$, a matrix can have many square roots; for example, $I_2$ has infinitely many.\n\n## Explanations\n\n### L1 human\n\nA positive semidefinite matrix has exactly one positive semidefinite square root.\nIn an orthonormal basis of eigenvectors, we take the square root of each eigenvalue.\nThis is like the positive square root of a nonnegative number.\n",
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