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<HTML>
<HEAD>
<TITLE>Matrix inverse</TITLE>
</HEAD>
<BODY bgcolor="#FFFFFF" fgcolor="#000000">
<P>
<font size="+3" color="green"><B>Matrix inverse</B></font></P>
<P>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%"><CODE>
x = INVERSE(m)</CODE>
</TD></TR>
</table></p>
<p>
The function <CODE>INVERSE(m)</code> returns the inverse
of the matrix <CODE>m</code>, which <i>must</i> be a
square matrix. The output is a matrix with the same shape as the argument. The
answer can be checked by using the inner product operator, for example:</p>
<p>
<font color="blue"><pre>
invm=INVERSE(m) ! find the inverse of m
=m<>invm ! this should be close to the identity matrix
</pre></font></p>
<p>
<font size="+1" color="green">Method</font></P>
<p>
Suppose that the matrix <i>A</i> has <i>n</i> rows and <i>n</i> columns.
Let <i>X</i> represent the inverse of <i>A</i>, and let <i>I</i> be the
identity matrix:</p>
<p>
<IMG ALIGN="top" SRC="inverseI01.png"></p>
<p>
The LU decomposition method is used for finding the inverse matrix <i>X</i>.
Write <i>A</i> as the product of two matrices: <i>A = L<>U</i> where
<i>L</i> is lower triangular and <i>U</i> is upper triangular. A lower
triangular matrix has elements only on the diagonal and below, while an upper
triangular matrix has elements only on the diagonal and above. This decomposition
is used to solve <i>n</i> sets of <i>n</i> linear equations. The matrix subscript
<i>*,j</i> represents the entire <i>j</i><sub>th</sub> column of that matrix.</p>
<p>
<IMG ALIGN="top" SRC="inverseI02.png"></p>
<p>
Solve for the <i>y</i> vectors, there will be <i>n</i> of them, such that
<i>L<>y = I<sub>*,j</sub></i> and then solve for the <i>j</i><sub>th</sub>
column of <i>X</i>:</p>
<p>
<i>U<>X<sub>*,j</sub> = y</i> for each <i>j=1,2,...,n</i></p>
<p>
Since <i>L</i> and <i>U</i> are triangular</p>
<p>
<IMG ALIGN="top" SRC="inverseI03.png"></p>
<p>
and</p>
<p>
<IMG ALIGN="top" SRC="inverseI04.png"></p>
</BODY>
</HTML>
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