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and factors it into a triangular matrix
times its transpose,
say
.
It is easy to reinvent the Cholesky factorization algorithm.
To do so,
simply write all the components of a 
triangular matrix
and then explicitly multiply these elements
times the transpose matrix 
.
You then find you have everything you need
to recursively build the elements of
from the elements of
.
Likewise,
for a 
matrix, etc.
The 
case shows that the Cholesky algorithm requires square roots.
Matrix elements are not always numbers.
Sometimes,
matrix elements are polynomials,
such as 
-transforms.
To avoid square roots,
there is a variation
of the Cholesky method.
In this variation, we factor
into
,
where 
is a diagonal matrix.
Once a matrix has been factored into upper and lower triangles,
solving simultaneous equations
is simply a matter of two back substitutions:
(We looked at a special case of back substitution
with Equation
(![[*]](icons/crossref.png){kind=icon}
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For example, we often encounter simultaneous equations of the form
.
Suppose the positive-definite matrix
has been factored into triangle form
.
To find
,
first backsolve
for the vector
.
Then,
we backsolve
.
When
happens to be a band matrix,
then the first back substitution is filtering down a helix,
and the second is filtering back up it.
Polynomial division is a special case of back substitution.
Poisson's equation
requires boundary conditions,
that we can honor when we filter starting from both ends.
We cannot simply solve Poisson's equation as
an initial-value problem.
We could insert the Laplace operator
into the polynomial division program,
but the solution would diverge.
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