complex array

complex_arrays.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Julia 複素行列について"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考1 https://en.wikibooks.org/wiki/Introducing_Julia/Arrays_and_tuples\n",
    "## 参考2 http://docs.julialang.org/en/release-0.4/stdlib/linalg/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. row vectorと1 x N matrixは異なる"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Complex{Int64},1}:\n",
       " 1+1im\n",
       " 1-2im\n",
       " 0+0im"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [1+1im,1-2im,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1+1im  1-2im  0+0im"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = [1+1im 1-2im 0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 上記のようにcommaをつけて定義すると、row vectorに、spaceで定義すると1 x N matrixとなる\n",
    "## 1 x N matrixを row vectorに変換したいとき"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1+1im  1-2im  0+0im"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[1,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 上ではだめで"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Complex{Int64},1}:\n",
       " 1+1im\n",
       " 1-2im\n",
       " 0+0im"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = transpose(b)[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a == c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### のようにしないとならない"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. 転置について\n",
    "### 転置には、単なる転置(transpose)と複素共役転置がある"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1+1im  1-2im  0+0im"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transpose(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1-1im  1+2im  0+0im"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ctranspose(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 紛らわしいのは省略記号がそれぞれ(.')と(')であることである"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1+1im  1-2im  0+0im"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x3 Array{Complex{Int64},2}:\n",
       " 1-1im  1+2im  0+0im"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 特に(')は実数行列では単なる転置なのでそれに慣れてしまうと、\n",
    "## 複素行列の時に複素共役をとる操作がはいっていることを忘れてしまうことである"
   ]
  }
 ],
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