Introduction to Matrices 29 th October 2014 Matrices

Introduction to Matrices 29 th October 2014

Matrices Definition: • A matrix is a rectangular array of numbers (or, in some cases, symbols). The numbers in the array are called the elements of the matrix. • The elements of a matrix are enclosed either in braces or in square brackets, as shown below.

Example • Let A denote the matrix • One may then write

• A matrix consists of horizontal rows of elements and vertical columns of elements. • The above matrix A has three horizontal rows and four vertical columns. We say, A is a 3 × 4 matrix, or A is of type 3 × 4. • An m × n matrix is a matrix with m rows and n columns:

• Each entry aij in the matrix is a number, where i tells what row the number is on, and j tells which column it is in. • For example, a 23 is the number in the second row and third column of the matrix.

• The number aij, which is in the ith row and jth column of A is referred as the i, jth element of A, or the (i, j) entry of A, and the matrix is denoted by

Definition • Two matrices A and B are equal iff they have the same size (equal dimensions) and all of their entries are equal, so aij = bij for all i, j. Definition • If we have an m × n matrix where m = n, then it is called a square matrix. For a square matrix, the entries a 11, a 22, . . . , ann are called the main diagonal or sometimes just the diagonal.

Example • The following are square matrices, where the first matrix is 3 × 3 and the second is 2× 2

Special Types of Matrices • Zero Matrix or Null Matrix is a matrix of any size in which every element is zero. • Examples: • Row matrix is a matrix with only one horizontal row. Example:

• A Column matrix is a matrix with only one vertical column. Example

• A diagonal matrix is a square matrix A=[aij] for which every term off the main diagonal is zero, i. e. , Example are diagonal matrices

• An identity matrix I is a diagonal matrix with all the diagonal elements having the value 1. • An identity matrix is a special square matrix with ones along the diagonal and zero elements elsewhere. • Examples

• A scalar matrix is a square diagonal matrix in which all the diagonal elements are equal. • Example: The following are scalar matrices;

• The Negative of a Matrix The negative of a given matrix A is a matrix denoted by -A whose elements are obtained by changing the sign of all the elements of matrix A. • Example: The negative of matrix

• Transpose of a Matrix The transpose of an m × n matrix A is an n × m matrix, denoted by AT. • The transpose of A is obtained by interchanging the rows and columns of A; that is, the first column of AT is the first row of A, the second column of AT is the second row of A, and so forth.

• Example: The following are some examples of matrices and their transposes.

• Symmetric Matrices A square matrix A=[aij] is said to be symmetric if it is identical to its transpose. In other words, A is symmetric if it is a square matrix for which aij = aji for all i, j. • Example: The matrix

• A Skew-Symmetrix Matrix A is a square matrix with the property that A = − AT. • A square matrix A=[aij] is said to be skew symmetric if aij = -aji for all i, j and all the elements of the main diagonal are zero.

• Example: The matrix is skew symmetric because

• Triangular Matrices A square matrix in which all the entries above the main diagonal are zero is called lower triangular, and a square matrix in which all the entries below the main diagonal are zero is called upper triangular. A matrix that is either upper triangular or lower triangular is called triangular.

• Example: The matrix is a general 3 x 3 upper triangular matrix, and the matrix is a general 3 x 3 lower triangular matrix.