ilyavasilev/numerical-methods-homeworks-2023
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Update Solutions_HW1.ipynb

Browse Source
This commit is contained in:
ilyavasilev 2023-10-05 23:32:35 +03:00
parent 89437dea68
commit a0e239491e
1 changed files with 2 additions and 2 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

4
HW1/Solutions_HW1.ipynb
View File

@ -132,7 +132,7 @@
"\n", "\n",
"$\\parallel UA \\parallel_{F} = \\sqrt{tr[(UA)^*(UA)]} = \\sqrt{tr[A^*U^*UA]}$.\n", "$\\parallel UA \\parallel_{F} = \\sqrt{tr[(UA)^*(UA)]} = \\sqrt{tr[A^*U^*UA]}$.\n",
"\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", "\n",
"$\\parallel UA \\parallel_{F} = \\sqrt{tr[A^*U^*UA]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$\n", "$\\parallel UA \\parallel_{F} = \\sqrt{tr[A^*U^*UA]} = \\sqrt{tr[A^*A]} = \\parallel A \\parallel_{F}$\n",
"\n", "\n",
@ -141,7 +141,7 @@
"$\\parallel AU \\parallel_{F} = \\sqrt{tr[(AU)^*(AU)]} = \\sqrt{tr[U^*A^*AU]}$.\n", "$\\parallel AU \\parallel_{F} = \\sqrt{tr[(AU)^*(AU)]} = \\sqrt{tr[U^*A^*AU]}$.\n",
"\n", "\n",
"Let's use the trace property $tr(AB) = tr(BA)$. So:\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", "\n",
"Thus $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$." "Thus $\\parallel UA \\parallel_{F} = \\parallel AU\\parallel_{F} = \\parallel A \\parallel_{F}$."
], ],