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Update Solutions_HW1.ipynb

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ilyavasilev 2023-10-05 23:32:35 +03:00
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HW1/Solutions_HW1.ipynb
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@ -132,7 +132,7 @@
"\n",
"$\\parallel UA \\parallel_{F} = \\sqrt{tr[(UA)^*(UA)]} = \\sqrt{tr[A^*U^*UA]}$.\n",
"\n",
"Let us take advantage of the fact that $U$ is unitary, i.e. $U^*U = I$. That:\n",
"Let us take advantage of the fact that $U$ is unitary, i.e. $U^*U = UU^*= I$. That:\n",
"\n",
"$\\parallel UA \\parallel_{F} = \\sqrt{tr[A^*U^*UA]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$\n",
"\n",
@ -141,7 +141,7 @@
"$\\parallel AU \\parallel_{F} = \\sqrt{tr[(AU)^*(AU)]} = \\sqrt{tr[U^*A^*AU]}$.\n",
"\n",
"Let's use the trace property $tr(AB) = tr(BA)$. So:\n",
"$\\parallel AU \\parallel_{F} = \\sqrt{tr[U^*A^*AU]} = \\sqrt{tr[A^*AUU^*]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$.\n",
"$\\parallel AU \\parallel_{F} = \\sqrt{tr[U^*A^*AU]} = \\sqrt{tr[AUU^*A^*]} = \\sqrt{tr[AA^*]} = \\parallel A \\parallel_{F}$.\n",
"\n",
"Thus $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$."
],