Matrix theory invariants

[Nico Van Cleemput]

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

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

also include:

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

note, if we need more invariants:

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