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June 9th, 2015, 02:43 AM   #1
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Joined: Oct 2014
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vector as product of matrices, exp function of vector

A vector (a11x, a22y, a33z) can be expended as:

\begin{pmatrix}
a_{11}x \\ a_{22}y\\ a_{33}z
\end{pmatrix}=
\begin{pmatrix}
a11 & a12 & a13\\
a21 & a22 & a23\\
a31 & a32 & a33
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0\\
0& 1 &0\\
0&0 & 1
\end{pmatrix}
\begin{pmatrix}
x\\y\\z
\end{pmatrix}

=

\begin{pmatrix}
* & * \\
* & * \\
* & *
\end{pmatrix}
\begin{pmatrix}
* & * & *\\
* & * & *
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0\\
0& 1 &0\\
0&0 & 1
\end{pmatrix}
\begin{pmatrix}
x\\y\\z
\end{pmatrix}

but how can I write the following vector as product of matrices:
\begin{pmatrix}
exp(a_{11})x \\ exp(a_{22})y\\ exp(a_{33})z
\end{pmatrix}
the annoying exp() prevents me doing any transformation!
Maybe the question is: is there a exp() function whose parameter is a vector instead of a single number?
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June 9th, 2015, 06:11 AM   #2
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" means . Is that really what you meant or did you mean ?

If it is the former, then you have just the matrix multiplication .

If it is the latter, it is a little more complicated. For a diagonal matrix, then .

If A is not diagonal, then you will have to try to "diagonalize" it. That is, find a matrix, B, such that where D is a diagonal matrix. In that case , of course, and .

(Only matrices that have a "full set of eigenvalues", that is, a basis for the vector space that are eigenvectors for the matrix, can be "diagonalized". For others, you have to use the "Jordan Normal Form" and that gets quite a bit more complicated.)
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