[2] The set of all m × n matrices over a field F denoted in this article Mmn(F) form a vector space.
, one legitimate basis set of matrices is: because any 2 × 2 matrix can be expressed as: where a, b, c,d are all real numbers.
This idea applies to other fields and matrices of higher dimensions.
Determinants are used for finding eigenvalues of matrices (see below), and for solving a system of linear equations (see Cramer's rule).
The eigenvalues are the roots of the characteristic polynomial: where I is the n × n identity matrix.
For all matrices A and B in Mmn(F), and all numbers α in F, a matrix norm, delimited by double vertical bars || ... ||, fulfills:[note 1] The Frobenius norm is analogous to the dot product of Euclidean vectors; multiply matrix elements entry-wise, add up the results, then take the positive square root: It is defined for matrices of any dimension (i.e. no restriction to square matrices).
Matrix elements are not restricted to constant numbers, they can be mathematical variables.