Saturday, May 22, 2010

HP 50G Determinants, Trace, Rank,Norms, Characteristic Polynomials, Eigen(Values|Vectors)

Let M be a matrix. In the HP 50G, various functions are available to obtain properties of the matrix.

 det(M) determinant trace(M) trace rank(M) rank cond(M) condition number abs(M) Frobenius norm, $$\sqrt{\sum x_{i,j}^2}$$ pcar(M) characyeristic polynomial if M is square. eigvl(M) eigenvalues of M egv(M) eigenvectors and eigenvaluesof M srad(M) spectral radius = abs. value of largest eigenvalue of M. rnrm(M) row norm is the maximum of all sums of the absolute values of elements in each row. snrm spectral norm of M, or largest singular value of M. cnrm column norm is the maximum of all sums of the absolute values of elements in each column.

Let us apply this to a random 3 x 3 matrix by calling the ranm function.
ranm({3,3}) sto M
This stored in my hp 50g the following matrix:

$\begin{array}{lll} -7 & 4 & -6 \\ -4 &-1 & 4 \\ 1 & 8 & 5 \\ \end{array}$

The outputs for the functions (except for egvl) is listed in the following:
• trace(M)
• -3
• rank(M)
• 3
• cond(M)
• 4.35304990758
• abs(M)
• 14.9666295471
• pcar(M)
• $$x^{3.} + 3.\cdot x^{2.} + -43., \cdot X - 541.$$
• egv(M)
• $$\{[[(1., 0),(1.,0), (-..250470319529,0.)] [(.217566565083,..5991855957363),(.2217566565093,-.5991757363),.50919544707,0.)] [(-.036364186761, -.456410356317),(-.036364186761,.456410356317),(1.,0.)]][(-5.911548611906, 5.135116596735), (-5.91154861906, -513516596735), (8.82309723813,0.)]\}$$
• 8.82309723813
• rnrm(M)
• 17
• snrm(M)
• 10.1517717328
• cnrm
• 15

Notes:

May 22

1. Wondering why the Latex plugin refuses to display the formula for an array.
2. How to split a long Latex output? Seems Wordpress is better in this regard.