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a0e239491e |
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HW1/PDFs/HW1_problem_6.pdf
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HW1/PDFs/HW1_problem_6.pdf
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HW1/PDFs/Solutions_HW1.pdf
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HW1/PDFs/Solutions_HW1.pdf
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HW1/PDFs/kmeans_solution.pdf
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HW1/PDFs/kmeans_solution.pdf
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@ -132,7 +132,7 @@
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"\n",
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"\n",
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"$\\parallel UA \\parallel_{F} = \\sqrt{tr[(UA)^*(UA)]} = \\sqrt{tr[A^*U^*UA]}$.\n",
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"$\\parallel UA \\parallel_{F} = \\sqrt{tr[(UA)^*(UA)]} = \\sqrt{tr[A^*U^*UA]}$.\n",
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"\n",
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"\n",
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"Let us take advantage of the fact that $U$ is unitary, i.e. $U^*U = I$. That:\n",
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"Let us take advantage of the fact that $U$ is unitary, i.e. $U^*U = UU^*= I$. That:\n",
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"\n",
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"\n",
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"$\\parallel UA \\parallel_{F} = \\sqrt{tr[A^*U^*UA]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$\n",
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"$\\parallel UA \\parallel_{F} = \\sqrt{tr[A^*U^*UA]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$\n",
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"\n",
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"\n",
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@ -141,7 +141,7 @@
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"$\\parallel AU \\parallel_{F} = \\sqrt{tr[(AU)^*(AU)]} = \\sqrt{tr[U^*A^*AU]}$.\n",
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"$\\parallel AU \\parallel_{F} = \\sqrt{tr[(AU)^*(AU)]} = \\sqrt{tr[U^*A^*AU]}$.\n",
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"\n",
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"\n",
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"Let's use the trace property $tr(AB) = tr(BA)$. So:\n",
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"Let's use the trace property $tr(AB) = tr(BA)$. So:\n",
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"$\\parallel AU \\parallel_{F} = \\sqrt{tr[U^*A^*AU]} = \\sqrt{tr[A^*AUU^*]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$.\n",
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"$\\parallel AU \\parallel_{F} = \\sqrt{tr[U^*A^*AU]} = \\sqrt{tr[AUU^*A^*]} = \\sqrt{tr[AA^*]} = \\parallel A \\parallel_{F}$.\n",
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"\n",
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"\n",
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"Thus $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$."
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"Thus $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$."
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],
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],
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