<s>
where	O
is	O
symmetric	B-Algorithm
positive-definite	B-Algorithm
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
can	O
be	O
derived	O
from	O
several	O
different	O
perspectives	O
,	O
including	O
specialization	O
of	O
the	O
conjugate	O
direction	O
method	O
for	O
optimization	O
,	O
and	O
variation	O
of	O
the	O
Arnoldi/Lanczos	O
iteration	O
for	O
eigenvalue	O
problems	O
.	O
</s>
<s>
Specific	O
choices	O
lead	O
to	O
various	O
methods	O
including	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
and	O
Gaussian	B-Algorithm
elimination	I-Algorithm
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
can	O
also	O
be	O
seen	O
as	O
a	O
variant	O
of	O
the	O
Arnoldi/Lanczos	O
iteration	O
applied	O
to	O
solving	O
linear	O
systems	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
for	O
,	O
is	O
found	O
by	O
Gram-Schmidt	B-Algorithm
orthogonalizing	I-Algorithm
against	O
followed	O
by	O
normalization	O
.	O
</s>
<s>
For	O
the	O
rest	O
of	O
discussion	O
,	O
we	O
assume	O
that	O
is	O
symmetric	B-Algorithm
positive-definite	B-Algorithm
.	O
</s>
<s>
With	O
symmetry	O
of	O
,	O
the	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
becomes	O
symmetric	B-Algorithm
and	O
thus	O
tridiagonal	O
.	O
</s>
<s>
Since	O
is	O
symmetric	B-Algorithm
positive-definite	B-Algorithm
,	O
so	O
is	O
.	O
</s>
<s>
is	O
symmetric	B-Algorithm
and	O
lower	O
triangular	O
simultaneously	O
and	O
thus	O
must	O
be	O
diagonal	O
.	O
</s>
