Matrix theory invariants

[nvcleemp]

here are some matrix invariants (from Horn and Johnson, 2nd ed):

• [x] trace(p.7)
• [x] determinant=det (p.8)
• [x] permanent=per (p.9)
• [x] rank (p.12) =number of non-zero-eigenvalues
• [x] maximum eigenvalue (p.45)
• [x] minimum eigenvalue
• [x] average eigenvalue
• [x] number of distinct eigenvalues (=cardinality of the spectrum)
• [x] spectral radius (p.52) =max of the absolute values of the eigenvalues
• [ ] maximum singular value (p.151) (any other singular value invariants, number of zeros, minimum, average, etc)
• [x] frobenius norm (p.321) or l_2 norm = sqrt(sum of squares of all the entries of the matrix)
• [x] l_1 norm (p.341) = sum of the absolute values of the entries of the matrix
• [x] l_inf norm (p.342) = max of the absolute value of the entries of the matrix
• [x] max column sum (p.344) = max of the sums of each column's entries = max row sum for symmetric matrices
• [x] ratio of the largest to smallest absolute eigenvalue (p.385)

also include:

• [x] number of rows (order of the square matrix)
• [x] nullity=number of zero-eigenvalues-number of rows - rank
• [x] separator=largest-second largest eigenvalue

note, if we need more invariants:

• [ ] any invariants of the adjoint (p.22) are invariants of the original matrix