Presentation on theme: "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."— Presentation transcript:
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.
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.