Matrices

Definitions  A matrix is an m x n array of scalars, arranged conceptually as m rows and n columns.  m is referred to as the row dimension  n is referred to as the column dimension  If m=n, the matrix is a square matrix.

Representations  Each element of array A is represented as:  Array A can thus be represented as:  The transpose of A is:  The column matrix of A is:  The corresponding row matrix is:

Matrix Operations  Scalar-matrix multiplication  Matrix-matrix addition: The sum makes sense only if the two matrices have the same dimensions.  Matrix-matrix multiplication The matrix-matrix product is defined only if the number of columns of A is the same as the number of rows of B.

Operation Properties

Row-Column Matrices vs. Transpose  We may represent any point in a space as a row or column matrix (or vector).  Transpose

Inverse  Matrix A is invertible if there exists a B such that: AB = I  Such matrix A is said to be nonsingular and B can denoted by A -1.  The inverse of a square matrix A exists if and only if |A|, determinant of A, is nonzero.

Identity Matrix  The identity matrix I is a square matrix with 1’s on the diagonal and 0’s elsewhere:  AI = A, IB = B

Rank The row (column) rank is the maximum number of linearly independent rows (columns).

Rank (II)  For an n x n matrix, if it is nonsingular, i.e., both of its row rank and column rank are n, the matrix has rank of n.

Basis Transformations

Cross Product