Section 2.2 More tools for constructing matrices
The number of coefficients in a linear system grows rapidly with its size: a system with 20 equations and 20 variables has a matrix with 400 coefficients. Entering these by hand would be tedious. We already have two ways to construct large matrices: zeros(m, n)
and ones(m, n)
but these are not invertible matrices. This section presents two other useful constructions.
The identity matrix is usually denoted \(I\) in mathematics, or \(I_n\) if it is necessary to emphasize its size. For example,
In Matlab this matrix would be created as eye(3)
. The name of the command was chosen because “eye” is pronounced the same as “I”. We also get scalars multiples of the identity matrix, for example 5*eye(3)
is
A diagonal matrix has zeros everywhere except on the main diagonal. The identity matrix \(I\) is a special case of a diagonal matrix. In Matlab, the command diag(v)
creates a diagonal matrix which has the elements of vector v
on its main diagonal. For example, diag([6 -4 7])
creates the matrix
Some problems in engineering and in differential equations lead to matrices where nonzero coefficients are close to the main diagonal but not exactly at it: for example,
To construct such matrices we can use diag(v, k)
which places the elements of vector v
on the diagonal parallel to the main one but \(k\) positions above it. So, diag(v, 1)
is just above the main diagonal and diag(v, -1)
is just below. The matrix shown above could be formed as
diag([2 2 2 2 2]) - diag([1 1 1 1], 1) - diag([1 1 1 1], -1)
This is an example of a tridiagonal matrix: there are only three diagonals with nonzero entries. More generally, a matrix is called sparse if most of its elements are 0. Large sparse matrices frequently arise in computations.
Example 2.2.1. Product of sparse matrices.
Let \(A\) be the 9×9 matrix with -1 on the main diagonal and 1 above it. Find and display the products of \(A\) with its transpose: \(A^TA\) and \(AA^T\text{.}\)
A = diag(ones(1, 8), 1) - eye(9); disp(A'*A) disp(A*A')
The two products are very similar but are not quite the same.