Function Axq; Function Qxa - HP 50g User Manual

This menu includes functions AXQ, CHOLESKY, GAUSS, QXA, and SYLVESTER.

Function AXQ

In RPN mode, function AXQ produces the quadratic form corresponding to a
matrix A in stack level 2 using the n variables in a vector placed in stack
n n
level 1. Function returns the quadratic form in stack level 1 and the vector of
variables in stack level 1. For example,
[[2,1,-1],[5,4,2],[3,5,-1]] `
returns
2: '2*X^2+(6*Y+2*Z)*X+4*Y^2+7*Z*y-Z^2'
1: ['X' 'Y' 'Z']

Function QXA

Function QXA takes as arguments a quadratic form in stack level 2 and a vector
of variables in stack level 1, returning the square matrix A from which the
quadratic form is derived in stack level 2, and the list of variables in stack level
1. For example,
returns
2: [[1 2 –8][2 1 0][-8 0 –1]]
1: ['X' 'Y' 'Z']
Diagonal representation of a quadratic form
Given a symmetric square matrix A, it is possible to "diagonalize" the matrix A
by finding an orthogonal matrix P such that P
matrix. If Q = x A x
the quadratic form Q so that it only contains square terms from a variable y,
['X','Y','Z'] ` XQ
'X^2+Y^2-Z^2+4*X*Y-16*X*Z' `
['X','Y','Z'] ` QX
T
is a quadratic form based on A, it is possible to write
T
A P = D, where D is a diagonal
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