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Function contained in this menu are: LQ, LU, QR,SCHUR, SVD, SVL.
Function LU
Function LU takes as input a square matrix
A
, and returns a lower-triangular
matrix
L
, an upper triangular matrix
U
, and a permutation matrix
P
, in stack
levels 3, 2, and 1, respectively. The results
L
,
U
, and
P
, satisfy the equation
P
⋅
A
=
L
⋅
U
. When you call the LU function, the calculator performs a Crout
LU decomposition of
A
using partial pivoting.
For example, in RPN mode:
[[-1,2,5][3,1,-2][7,6,5]] LU
produces:
3:[[7 0 0][-1 2.86 0][3 –1.57 –1]
2: [[1 0.86 0.71][0 1 2][0 0 1]]
1: [[0 0 1][1 0 0][0 1 0]]
In ALG mode, the same exercise will be shown as follows:
Orthogonal matrices and singular value decomposition
A square matrix is said to be orthogonal if its columns represent unit vectors
that are mutually orthogonal. Thus, if we let matrix
U
= [
v
1
v
2
…
v
n
] where
the
v
i
, i = 1, 2, …, n, are column vectors, and if
v
i
•
v
j
=
δ
ij
, where
δ
ij
is the
Kronecker’s delta function, then
U
will be an orthogonal matrix. This
conditions also imply that
U
⋅
U
T
=
I
.