Function Lu; Orthogonal Matrices And Singular Value Decomposition; Function Svd - HP 50g User Manual

Graphing calculator.

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:
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
i = 1, 2, ..., n, are column vectors, and if v
delta function, then U will be an orthogonal matrix. This conditions also imply
T
= I.
that U U
The Singular Value Decomposition (SVD) of a rectangular matrix A
determining the matrices U, S, and V, such that A
where U and V are orthogonal matrices, and S is a diagonal matrix. The
diagonal elements of S are called the singular values of A and are usually
ordered so that s
V are the corresponding singular vectors.

Function SVD

In RPN, function SVD (Singular Value Decomposition) takes as input a matrix
A
, and returns the matrices U
n m
and 1, respectively. The dimension of vector s is equal to the minimum of the
values n and m. The matrices U and V are as defined earlier for singular value
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]]
s
, for i = 1, 2, ..., n-1. The columns [u
i
i+1
v
=
i
j
m n
, V
, and a vector s in stack levels 3, 2,
n n
m m
v
... v
] where the v
1
2
n
, where
is the Kronecker's
ij
ij
m n
= U
S
m m
m n
] of U and [v
j
,
i
consists in
T
V
,
n n
] of
j
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