6.44.4  Creating a matrix with a formula or function: makemat matrix

You can use a function or a formula to specify the elements of a matrix with the makemat or matrix command.

• makemat takes three arguments:
  • f, a function of two variables j and k which returns the value of a_j,k, the element at row index j and column index k of the resulting matrix.
  • n and p, two positive integers.
• makemat(f,n,p) returns the n× p matrix A=(a_j,k) with a_j,k=f(j,k) for j=1..n and k=1..p.

Example.
Input:

or:

Output:

Note that the indices are counted starting from 0.

The matrix command can be used similarly, but note that the arguments are given in a different order and the indices start at 1.
(matrix can also be used to turn tables into matrices; see Section 6.46.1.)

• matrix takes two mandatory arguments and one optional argument:
  • n and p, two integers.
  • Optionally, f, a function of two variables j and k which should return the value of a_j,k, the element at row index j and column index k of the resulting matrix.
• matrix(n,p) returns the n× p matrix consisting of all 0s.
• matrix(n,p,f) returns the n× p matrix A=(a_j,k) with a_j,k=f(j,k) for j=1..n and k=1..p.

Examples.

• Input:
  matrix(2,3)
  Output:

  ⎛ ⎜ ⎝ 0 0 0 0 0 0 ⎞ ⎟ ⎠

• Input:
  matrix(4,3,(j,k)->j+k)
  or:

  h(j,k):=j+k

  matrix(4,3,h)

  Output:

  ⎛ ⎜ ⎜ ⎜ ⎝ 0 1 2 1 2 3 2 3 4 3 4 5 ⎞ ⎟ ⎟ ⎟ ⎠