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Version:
4.4.5.dfsg-3ubuntu2 ▾
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<HTML>
<HEAD>
<TITLE>INNER PRODUCT</TITLE>
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<BODY bgcolor="#FFFFFF" fgcolor="#000000">
<P><A NAME="inner product"></A>
<font size="+3" color="green"><B>INNER PRODUCT</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>
vout = v1 <> v2</CODE>
</TD></TR>
</table></p>
<p>
The inner product operator, <code><></code>, operating on two vectors
produces a scalar; operating on a vector and a matrix produces a vector; and
operating on two matrices produces a matrix.</p>
<P><font size=+2 color="green">Operating on two vectors</font></p>
<P>
The inner product operating on two vectors produces a scalar, whose value
is equal to the sum of the products of the vectors' elements. The two
vectors <EM>must</EM> be the same length. Suppose <code>X</code> and <code>Y</code>
are vectors of length N. Then <code>x<>y = x[1]*y[1] + x[2]y[2] + ... + x[N]*y[N]</code>.</p>
<P>
<font size="+1" color="green">Example</font></p>
<P>
Suppose you have two vectors: <code>X = [1;3;5]</CODE> and <CODE>Y = [2;4;6]</code></p>
<P>
Then: <code>X<>Y = 1*2 + 3*4 + 5*6 = 44</CODE></p>
<P>
<font size=+2 color="green">Operating on a vector and a matrix</font></p>
<P>
The inner product operating on a vector and a matrix produces a
vector. If the vector is the first operand, its length <EM>must</EM> be equal to
the number of rows of the matrix. The resultant vector length will be the
number of columns of the second operand matrix.</p>
<p>
If <code>X</code> is a vector of length N, and <code>A</code> is a matrix with N rows and M columns,
then <code>X<>A</code> is a vector where
<code>(X<>A)[i] = X[1]*A[1,i] + X[2]*A[2,i] + ... + X[N]*A[N,i]</code>. The vector
<code>X<>A</code> will have M elements.</p>
<P>
<font size="+1" color="green">Example</font></p>
<P>
The inner product of a vector and a matrix:</p>
<P>
<code>X = [1;3;5]</CODE> and
<pre>
| 1 4 |
M = | 2 5 |
| 3 6 |
</pre></p>
<P>
Then: <code>X<>M = [22;49]</CODE></p>
<P>
<font size="+2" color="green">Operating on a matrix and a vector</font></p>
<P>
The inner product operating on a matrix and a vector produces a vector. If
the vector is the second operand, its length <EM>must</EM> be equal to the
number of columns of the matrix, and the resultant vector length will be the
number of rows of the first operand matrix.</p>
<p>
If <code>A</code> is a matrix with N rows and M columns, and <code>X</code> is a vector of length M,
then <code>A<>X</code> is a vector where
<code>(A<>X)[i] = A[i,1]*X[1] + A[i,2]*X[2] + ... + A[i,M]*X[M]</code>. The vector
<code>X<>A</code> will have N elements.</p>
<P>
<font size="+1" color="green">Example</font></p>
<P>
The inner product of a matrix and a vector:</p>
<P>
<code>X = [1;3;5]</code>
<pre>
M = | 1 3 5 |
| 2 4 6 |
</pre></p>
<P>
Then: <code>M<>X = [35;44]</CODE></p>
<P>
<font size="+2" color="green">Operating on two matrices</font></p>
<P>
The inner product operating on two matrices produces a matrix.
The number of columns of the first operand matrix <EM>must</EM> be equal to the
number of rows of the second operand matrix. The resultant matrix will be a
square matrix with the number of rows and the number of columns equal to the
number of rows of the first operand.</P>
<P>
If <code>A</code> is an N by M matrix and <code>B</code> is an M by N matrix,
then <CODE>A<>B</code> is an N by N matrix, where <code>(A<>B)[i,j] =
A[i,1]*B[1,j] + A[i,2]*B[2,j] + ... + A[i,M]*B[M,j]</code> for
<CODE>i = [1;2;...;N]</CODE> and <CODE>j = [1;2;...;N]</CODE></p>
<P>
<font size="+1" color="green">Example</font></p>
<P>
The inner product of two matrices:</p>
<P>
Let
<pre>
| 11 12 13 |
A = | 21 22 23 |
| 31 32 33 |
</pre>
and
<pre>
| 1 2 3 |
B = | 2 4 6 |
| 3 6 9 |
</pre>
then
<pre>
| 74 148 222 |
A<>B = | 134 268 402 |
| 194 388 582 |
</pre></p>
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