added problem 5 to pset 3

psets/pset3.ipynb

@@ -134,12 +134,30 @@
     "@show svdvals(S)[1:6]"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "b9415ba3",
+   "metadata": {},
+   "source": [
+    "## Problem 5 (6 points)\n",
+    "\n",
+    "Suppose that $X$ is a random $m \\times n$ matrix (drawn from some probability distribution) and its mean (expectation value) is $E[X] = X_0$.\n",
+    "\n",
+    "Derive the following identity for the variance (which we used in class):\n",
+    "\n",
+    "$$\n",
+    "E[\\Vert X - X_0 \\Vert_F^2] = E[\\Vert X \\Vert_F^2] - \\Vert X_0 \\Vert_F^2\n",
+    "$$\n",
+    "\n",
+    "(If you just write out the Frobenius norm definition $\\Vert A \\Vert_F^2 = \\text{tr}[A^T A]$, the problem should be very straightforward.  Remember that the expectation value is conceptually just a sum, so it can be moved inside any linear operation like the trace.)"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "88ab5b5e",
    "metadata": {},
    "source": [
-    "## Problems 5, 6, etc: coming soon"
+    "## Problems 6, etc: coming soon"
    ]
   },
   {