Matrix Definitions It is assumed you are already familiar with the terms matrix, matrix transpose, vector, row vector, column vector, unit vector, zero matrix, diagonal matrix, identity matrix You should also be familiar with the operations matrix addition, matrix negation and subtraction, and matrix multiplication, which is associative but not commutative

More Definitions

Matrix Inverses If the following matrices are nonsingular, then ( B A ) -1 = A -1 B -1 and ( A -1 ) T = ( A T ) -1 Vectors x1, … , xn are linearly dependent if there exist coefficients c1, … , cn all not zero such that c1x1 + … + cnxn = 0; otherwise the vectors are linearly independent

Rank of a Matrix The column rank of a nonzero m x n matrix is the largest set of linearly independent columns The row rank is similar; the row rank equals the column rank which equals the rank of the matrix A square n x n matrix has full rank if equal to n

Determinants

Determinant Properties

Positive-definite Matrices An n x n matrix has full column rank if its rank is n