Function Lq; Function Qr; Matrix Quadratic Forms - HP 49g+ User Manual

Graphing calculator.

1: [[-1.03 1.02 3.86 ][ 0 5.52 8.23 ][ 0 –1.82 5.52]]

Function LQ

The LQ function produces the LQ factorization of a matrix A
lower L
trapezoidal matrix, a Q
×
n
m
permutation matrix, in stack levels 3, 2, and 1. The matrices A, L, Q and P
are related by P⋅A = L⋅Q. (A trapezoidal matrix out of an n×m matrix is the
equivalent of a triangular matrix out of an n×n matrix). For example,
[[ 1, -2, 1][ 2, 1, -2][ 5, -2, 1]] LQ
produces
3: [[-5.48 0 0][-1.10 –2.79 0][-1.83 1.43 0.78]]
2:
[[-0.91 0.37 -0.18] [-0.36 -0.50 0.79] [-0.20 -0.78 -0.59]]
1: [[0 0 1][0 1 0][1 0 0]]

Function QR

In RPN, function QR produces the QR factorization of a matrix A
a Q
orthogonal matrix, a R
×
n
n
permutation matrix, in stack levels 3, 2, and 1. The matrices A, P, Q and R
are related by A⋅P = Q⋅R. For example,
[[ 1,-2,1][ 2,1,-2][ 5,-2,1]] QR
produces
3: [[-0.18 0.39 0.90][-0.37 –0.88 0.30][-0.91 0.28 –0.30]]
2: [[ -5.48 –0.37 1.83][ 0 2.42 –2.20][0 0 –0.90]]
1: [[1 0 0][0 0 1][0 1 0]]
Note: Examples and definitions for all functions in this menu are available
through the help facility in the calculator. Try these exercises in ALG mode to
see the results in that mode.

A quadratic form from a square matrix A is a polynomial expression
originated from x⋅A⋅x
T
. For example, if we use A = [[2,1,–1][5,4,2][3,5,–
1]], and x = [X Y Z]
T
, the corresponding quadratic form is calculated as
orthogonal matrix, and a P
×
m
m
upper trapezoidal matrix, and a P
×
n
m
returning a
×
n
m
×
n
n
returning
×
n
m
×
m
m
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