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  "body": "\n## Definition\n\nLet $A = (a_{ij})$ be an $m \\times n$ matrix with complex entries.\nThe *conjugate transpose* of $A$ is the $n \\times m$ matrix $A^*$ with entries\n$$(A^*)_{ij} = \\overline{a_{ji}}$$\nfor $1 \\le i \\le n$ and $1 \\le j \\le m$, where $\\overline{z}$ is the complex conjugate of $z$.\n\n## Remarks\n\nOther notations are $A^H$ and $A^\\dagger$.\nFor a real matrix, the conjugate transpose is the [[transpose|09DXWYRD43]].\nThe rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold.\nFor the standard inner product on $\\C^n$, $\\langle Ax, y \\rangle = \\langle x, A^* y \\rangle$.\n\n## Explanations\n\n### L1 human\n\nThe conjugate transpose flips a complex matrix over its diagonal and conjugates each entry.\nFor example, the conjugate transpose of $\\begin{pmatrix} 1 \u0026 i \\\\ 0 \u0026 2 \\end{pmatrix}$ is $\\begin{pmatrix} 1 \u0026 0 \\\\ -i \u0026 2 \\end{pmatrix}$.\nFor complex matrices it plays the role that the transpose plays for real matrices.\n",
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