{ "metadata": { "name": "REF_linear_algebra" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "

# Trace Properties

\n", "

For the square matric A, define tr A = trace(A) = $\\sum_{i}A_{i,i}$. Let $\\bigtriangledown f(A)$ be the matrix of partial\n", " derivatives of f with respect to the elements of A. The following hold:\n", "

\n", "
1. tr AB = tr BA
2. \n", "
3. $\\bigtriangledown tr AB = B^T$
4. \n", "
5. $tr A = tr A^T$
6. \n", "
7. $\\bigtriangledown tr ABA^TC = CAB + C^TAB^T$
8. \n", " \n", "
\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

# Transpose Properties

\n", "
\n", "
1. $(A+B)^T= A^T+ B^T$
2. \n", "
3. $(AB)^T = B^TA^T$
4. \n", "
5. $det(A^T) = det(A)$
6. \n", "
7. $(A^T)^{-1}=(A^{-1})^T$
8. \n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

# Inverse Properpties

\n", "\n", "1. $(AB)^{-1} = B^{-1}A^{-1}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

# Matrix Derivatives

\n", "\n", "1. $\\frac{\\partial}{\\partial \\mathbf{x}} (\\mathbf{x}^T \\mathbf{y}) = \\frac{\\partial}{\\partial \\mathbf{x}} (\\mathbf{y}^T \\mathbf{x}) = \\mathbf{y}$" ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }