> Would anybody have a version of this in EQN. I do not mean the > simple 2x2 matrix but the generic nxn square matrix
This is fairly straightforward with a few suitable eqn
definitions to simplify entering the equation. The attached
PDF was created with the following code and my own paragraph
and section macros. (Use "left [" and "right ]" instead of
my bracket macro if you want to use the equation with some
other macro package.) Note that since I've entered the matrix
column-wise it appears transposed in the text editor display.
$
define ... % roman " . . . " %
define vdots % roman "\v'-.6m'\z.\v'.5m'\z.\v'.5m'.\v'-.4m'" %
define ddots % roman "\v'-.6m' .\v'.5m' .\v'.5m' .\v'-.4m' " %
define Gjj % G sub back 30 jj %
define Gji % G sub back 30 ji %
define Gii % G sub back 10 ii %
define Gij % G sub back 10 ij %
define Gkk % G sub back 10 kk %
define | % above %
$
.SE
Givens rotations
.PP
A Givens rotation is represented by a matrix of the form
.EQ
set column_sep 20
G ( i, j, theta ) =
bracket { matrix {
ccol { 1 | vdots | 0 | vdots | 0 | vdots | 0 }
ccol { ... | ddots | ... | "" | ... | "" | ... }
ccol { 0 | vdots | c | vdots | s | vdots | 0 }
ccol { ... | "" | ... | ddots | ... | "" | ... }
ccol { 0 | vdots | -s | vdots | c | vdots | 0 }
ccol { ... | "" | ... | "" | ... | ddots | ... }
ccol { 0 | vdots | 0 | vdots | 0 | vdots | 1 }
} } ,
.EN
which is, when $i > j$,
.EQ
set column_sep 100
Gkk = 1 roman " for " k != i, j , fwd 100
Gii = Gjj = cos ( theta ) , fwd 100
Gji = - Gij = - sin ( theta ) .
.EN
(See https://en.wikipedia.org/wiki/Givens_rotation.)
givensrotation.pdf
Description: Adobe PDF document
