| 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) $$ \{[[(1., 0),(1.,0), (-..250470319529,0.)] [(.217566565083,..5991855957363),(.2217566565093,-.5991757363),.50919544707,0.)] [(-.036364186761, -.456410356317),(-.036364186761,.456410356317),(1.,0.)]][(-5.911548611906, 5.135116596735), (-5.91154861906, -513516596735), (8.82309723813,0.)]\}$$
- srad(M) 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.
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