Vectors and Matrices

$M_{\mathrm{HC}}\mathrm{is}\mathrm{our}\mathrm{matrix}\mathrm{that}\mathrm{maps}\mathrm{the}\mathrm{first}\mathrm{three}\mathrm{hydrocarbons}\mathrm{to}\mathrm{number}\mathrm{of}\mathrm{hydrogens}\mathrm{and}\mathrm{carbons}$

In[1]:=

Out[1]=

$\left\{\left\{4,1\right\},\left\{8,3\right\},\left\{10,4\right\}\right\}$

Out[2]//MatrixForm=

$\left(\begin{array}{ll}4& 1\\ 8& 3\\ 10& 4\end{array}\right)$

$\mathrm{PTmatrix}\mathrm{is}\mathrm{our}\mathrm{matrix}\mathrm{of}\mathrm{kinetic}\mathrm{data}\mathrm{that}\mathrm{gives}\mathrm{rates}\mathrm{of}\mathrm{change}\mathrm{of}a\mathrm{particular}\mathrm{atomic}\mathrm{species}\left(C\mathrm{or}H\right)\mathrm{as}a\mathrm{function}\mathrm{of}\mathrm{pressure}\mathrm{and}\mathrm{temperature}\left(\mathrm{see}\mathrm{lecture}\mathrm{notes}\mathrm{corresponding}\mathrm{to}\mathrm{this}\mathrm{Mathematica}\mathrm{notebook}\right).$

In[3]:=

Out[4]//MatrixForm=

$\left(\begin{array}{ll}\alpha & \beta \\ \gamma & \delta \\ ϵ& \varphi \end{array}\right)$

In[5]:=

$\mathrm{MPT}=M_{\mathrm{HC}}.\mathrm{PTmatrix}$

$\mathrm{Dot}\text{::}\mathrm{dotsh}:\text{Tensors}\left\{\left\{4,1\right\},\left\{8,3\right\},\left\{10,4\right\}\right\}\text{and}\left\{\left\{\alpha ,\beta \right\},\left\{\gamma ,\delta \right\},\left\{ϵ,\varphi \right\}\right\}\text{have incompatible shapes.}\mathrm{More\dots }$

Out[5]=

$\left\{\left\{4,1\right\},\left\{8,3\right\},\left\{10,4\right\}\right\}.\left\{\left\{\alpha ,\beta \right\},\left\{\gamma ,\delta \right\},\left\{ϵ,\varphi \right\}\right\}$

The matrix multiplication does not work because the sizes are inconsistent.

In[6]:=

$\mathrm{Clear}\left[\mathrm{MPT}\right]$

In[7]:=

Out[8]//MatrixForm=

$\left(\begin{array}{ll}4\alpha +8\gamma +10ϵ& 4\beta +8\delta +10\varphi \\ \alpha +3\gamma +4ϵ& \beta +3\delta +4\varphi \end{array}\right)$

In[9]:=

Out[10]//MatrixForm=

$\left(\begin{array}{ll}-\frac{\beta +3\delta +4\varphi }{2\left(2\beta \gamma -2\alpha \delta +3\beta ϵ+\delta ϵ-3\alpha \varphi -\gamma \varphi \right)}& \frac{2\beta +4\delta +5\varphi }{2\beta \gamma -2\alpha \delta +3\beta ϵ+\delta ϵ-3\alpha \varphi -\gamma \varphi }\\ -\frac{\alpha +3\gamma +4ϵ}{2\left(-2\beta \gamma +2\alpha \delta -3\beta ϵ-\delta ϵ+3\alpha \varphi +\gamma \varphi \right)}& \frac{2\alpha +4\gamma +5ϵ}{-2\beta \gamma +2\alpha \delta -3\beta ϵ-\delta ϵ+3\alpha \varphi +\gamma \varphi }\end{array}\right)$

In[11]:=

$\mathrm{Det}\left[\mathrm{MPT}\right]$

Out[11]=

$-4\beta \gamma +4\alpha \delta -6\beta ϵ-2\delta ϵ+6\alpha \varphi +2\gamma \varphi$

In[12]:=

$\mathrm{RowReduce}\left[M_{\mathrm{HC}}\right]//\mathrm{MatrixForm}$

Out[12]//MatrixForm=

$\left(\begin{array}{ll}1& 0\\ 0& 1\\ 0& 0\end{array}\right)$

Same example for water and water complexes: use the matrix watmat to store molecular formulas for each type of molecule in the system

In[13]:=

Out[14]//MatrixForm=

$\left(\begin{array}{ll}2& 4\\ 1& 2\end{array}\right)$

The vector molvec is used to store the number of each kind of molecule

In[15]:=

$\mathrm{molvec}=\left\{\mathrm{h20},\mathrm{h402}\right\}$

Out[15]=

$\left\{\mathrm{h20},\mathrm{h402}\right\}$

The vector atomvec is used to store the number of each atomic species that is present

In[16]:=

$\mathrm{atomvec}=\left\{h,o\right\}$

Out[16]=

$\left\{h,o\right\}$

In[17]:=

$\mathrm{atomvec}//\mathrm{MatrixForm}$

Out[17]//MatrixForm=

$\left(\text{⁠}\begin{array}{l}h\\ o\end{array}\text{⁠}\right)$

The vector eq is now defined and its two elements are equations that give the number of hydrogen atoms and the number of oxygen atoms:

In[18]:=

$\mathrm{eq}\left[1\right]=\left(\mathrm{watmat}.\mathrm{molvec}\right)\left[\left[1\right]\right]⩵\mathrm{atomvec}\left[\left[1\right]\right]$

Out[18]=

$2\mathrm{h20}+4\mathrm{h402}⩵h$

In[19]:=

$\mathrm{eq}\left[2\right]=\left(\mathrm{watmat}.\mathrm{molvec}\right)\left[\left[2\right]\right]⩵\mathrm{atomvec}\left[\left[2\right]\right]$

Out[19]=

$\mathrm{h20}+2\mathrm{h402}⩵o$

In[20]:=

$?\mathrm{Eliminate}$

$\text{Eliminate[eqns, vars] eliminates variables between a set of simultaneous equations.}\mathrm{More\dots }$

In[21]:=

$\mathrm{Eliminate}\left[\left\{\mathrm{eq}\left[1\right],\mathrm{eq}\left[2\right]\right\},\mathrm{molvec}\right]$

Out[21]=

$2o⩵h$

Created by Mathematica  (September 21, 2005)