From a0e239491ee8f1d6f27a0d0c8cef4621348813a7 Mon Sep 17 00:00:00 2001 From: ilyavasilev Date: Thu, 5 Oct 2023 23:32:35 +0300 Subject: [PATCH] Update Solutions_HW1.ipynb --- HW1/Solutions_HW1.ipynb | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/HW1/Solutions_HW1.ipynb b/HW1/Solutions_HW1.ipynb index 1a96017..ff57aff 100755 --- a/HW1/Solutions_HW1.ipynb +++ b/HW1/Solutions_HW1.ipynb @@ -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 $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$." ],