Matrices

Definition of a matrix

A matrix is a two-dimensional array. The dimensions of a matrix are given by its number of rows n and columns m. We designate a matrix by a bold letter. The matrix A is A. The dimensions of a are n x m (n by m). The elements of a are a_ij. The index i is the row index and the index j is the column index. We represent A as:

Special cases

A square matrix has n = m.

A diagonal matrix D has values only along the diagonal d₁₁, d₂₂, d₃₃, etc. and zero elsewhere.

The unit matrix or identity matrix I has ones along the diagonal.

A column vector is a matrix of dimension n x 1.

A row vector is a matrix of dimension 1 x n.

Matrix addition and subtraction

Two matrices can be added or subtracted only if they have the same dimesions. The result of adding matrices A and B is obtained by adding their corresponding elements. A + B = C is obtained by adding a_ij + b_ij = c_ij for all i and j.

Matrix multiplication

In order to multiply two matrices AB = C the number of columns in the A matrix must be the same as the number of rows in the B matrix. If A is a k x l matrix and B is a l x m matrix, then C has dimensions k x m. The prescription for matrix multiplication is:

The first row elements of the A matrix are multiplied by the first column elements of the B matrix to give the first element of the C matrix.

Note that the above equation indicates the sum of the products of the elements c₁₁ = a₁₁b₁₁ + a₁₂b₂₁ + a₁₃b₃₁ + … until the row and column are finished.

The c₂₁ element is obtained in a similar fashion:

This process is continued until all of the elements of the C matrix have been obtained.

The order of operation is important. This can be simply illustrated with the following important example.

The product of a row matrix and a column matrix is a scalar.

The product of a n x 1 column matrix and a 1 x n row matrix is a n x n matrix.

Matrix Transpose

The transpose of a matrix is obtained by interchanging the rows and columns of a matrix.

Matrix Inverse

The inverse of a matrix is defined such that AA^-1 = A^-1 A = I where I is the identity matrix.

The determinant of a matrix

The determinant of a matrix is a number obtained by summing n! products of a n x n matrix. This is best illustrated by starting with a 2 x 2 matrix. The determinant of a 2 x 2 matrix is:

For larger dimension matrices, the determinant can be obtained by reduction to lower-order determinants (minors). This is illustrated for a 3 x 3 matrix.

The 2 x 2 determinant is obtained as above for each of the minors.

The determinant of A is obtained by summing across any row as follows:

Matrix solution of linear equations

Matrices are important tools for the solution of linear equations. The general form of a set of simultaneous linear equations is

Ax = b. This illustrated below:

Here the x_i are the unknowns and a_ij are the linear coefficients. To find the ith coefficient x_i we replace that column of the A matrix by the b vector.

The unknown is given by the ratio of the determinants,