<s>
In	O
linear	B-Language
algebra	I-Language
,	O
the	O
modal	B-Algorithm
matrix	I-Algorithm
is	O
used	O
in	O
the	O
diagonalization	B-Algorithm
process	I-Algorithm
involving	O
eigenvalues	O
and	O
eigenvectors	O
.	O
</s>
<s>
Specifically	O
the	O
modal	B-Algorithm
matrix	I-Algorithm
for	O
the	O
matrix	O
is	O
the	O
n	O
×	O
n	O
matrix	O
formed	O
with	O
the	O
eigenvectors	O
of	O
as	O
columns	O
in	O
.	O
</s>
<s>
where	O
is	O
an	O
n	O
×	O
n	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
the	O
eigenvalues	O
of	O
on	O
the	O
main	O
diagonal	O
of	O
and	O
zeros	O
elsewhere	O
.	O
</s>
<s>
The	O
matrix	O
is	O
called	O
the	O
spectral	B-Algorithm
matrix	I-Algorithm
for	O
.	O
</s>
<s>
A	O
generalized	O
modal	B-Algorithm
matrix	I-Algorithm
for	O
is	O
an	O
n	O
×	O
n	O
matrix	O
whose	O
columns	O
,	O
considered	O
as	O
vectors	O
,	O
form	O
a	O
canonical	O
basis	O
for	O
and	O
appear	O
in	O
according	O
to	O
the	O
following	O
rules	O
:	O
</s>
<s>
This	O
example	O
illustrates	O
a	O
generalized	O
modal	B-Algorithm
matrix	I-Algorithm
with	O
four	O
Jordan	O
chains	O
.	O
</s>
<s>
An	O
"	O
almost	O
diagonal	O
"	O
matrix	O
in	O
Jordan	O
normal	O
form	O
,	O
similar	B-Algorithm
to	O
is	O
obtained	O
as	O
follows	O
:	O
</s>
<s>
where	O
is	O
a	O
generalized	O
modal	B-Algorithm
matrix	I-Algorithm
for	O
,	O
the	O
columns	O
of	O
are	O
a	O
canonical	O
basis	O
for	O
,	O
and	O
.	O
</s>
