Fixed math for P = FPF^ #274

Author: rlabbe

06-Multivariate-Kalman-Filters.ipynb

@@ -1769,12 +1769,11 @@
     "If you have some experience with linear algebra and statistics, this may help. The covariance due to the prediction can be modeled as the expected value of the error in the prediction step, given by this equation. \n",
     "\n",
     "$$\\begin{aligned}\n",
-    "\\bar{\\mathbf P} &= \\mathbb E[(\\mathbf{Fx})(\\mathbf{Fx})^\\mathsf T]\\\\\n",
-    " &= \\mathbb E[\\mathbf{Fxx}^\\mathsf T\\mathbf F^\\mathsf T] \\\\\n",
-    " &= \\mathbf F\\, \\mathbb E[\\mathbf{xx}^\\mathsf T]\\, \\mathbf F^\\mathsf T\n",
+    "\\bar{\\mathbf P} &= \\mathbb E[(\\mathbf{Fx - \\bar \\mu})(\\mathbf{Fx - \\bar\\mu})^\\mathsf T]\\\\\n",
+    " &= \\mathbf F\\, \\mathbb E[\\mathbf{(x- \\bar\\mu)(x- \\bar\\mu)}^\\mathsf T]\\, \\mathbf F^\\mathsf T\n",
     "\\end{aligned}$$\n",
     "\n",
-    "Of course, $\\mathbb E[\\mathbf{xx}^\\mathsf T]$ is just $\\mathbf P$, giving us\n",
+    "Of course, $\\mathbb E[\\mathbf{(x- \\bar\\mu)(x- \\bar\\mu)}^\\mathsf T]$ is just $\\mathbf P$, giving us\n",
     "\n",
     "$$\\bar{\\mathbf P} = \\mathbf{FPF}^\\mathsf T$$\n",
     "\n",