q×q

q×q denotes a square matrix with q rows and q columns. The symbol q represents a positive integer, and a q×q matrix A has entries aij for i = 1,…,q and j = 1,…,q. The set of all q×q matrices over a field F is commonly denoted Mq(F).

As a linear algebra object, a q×q matrix represents a linear transformation from the q-dimensional vector space Fq to itself. Matrix operations preserve dimension, so the product of two q×q matrices is again a q×q matrix. The identity matrix Iq, with ones on the diagonal and zeros elsewhere, acts as the identity element for multiplication.

Key properties and concepts associated with q×q matrices include the determinant and invertibility. The determinant det(A) is defined for any q×q matrix and is zero precisely when A is not invertible. If det(A) ≠ 0, A has a unique inverse A−1 such that AA−1 = A−1A = Iq. Eigenvalues are roots of the characteristic equation det(A − λIq) = 0, and they reveal fundamental aspects of the transformation represented by A. The trace, the sum of diagonal entries, and the rank, the dimension of the column space, are also central invariants.

Applications span solving systems of q equations in q variables, studying geometric transformations in q-dimensional space, and performing numerical methods, such as eigenvalue decompositions and matrix factorizations, which underpin many scientific and engineering computations. In contexts involving finite fields, a q×q matrix may have entries from GF(q), the finite field with q elements.