5.4 The n × n Helmert matrix H, n ≥ 2, is an orthogonal matrix whose rows are defined as follows:

● For j = 1, ..., n − 1, the j row is given by

(1, ..., 1,−j, 0, ..., 0)∕√j(j + 1), where 1 is repeated j times and 0 is repeated n − j − 1 times.

● The nth row is given by (1, ..., 1)∕√n.

That is, for n = 3 H =

⎡

⎢

⎢

⎢

⎣

1∕

√

2 −1∕

√

2 0 1∕

√

6 1∕

√

6 −2∕

√

6 1∕

√

3 1∕

√

3 1∕

√

3

⎤

⎥

⎥

⎥

⎦

(a) Show that the rows of H are orthonormal; that is, they have unit norm and they are orthogonal to each other. Show that in matrix form, these statements take the form HHT = In, that is, H is an orthogonal matrix.

(b) The sub-Helmert matrix, denoted H∖n, say, is defined as the first n − 1 rows of H. Show that H∖n is “row-orthonormal,” that is, H∖nHT

∖n = In−1.

Show that each row of H∖n is orthogonal to the constant vector ????n, H∖n????n = ????n−1. Thus, if y is an n-dimensional vector and z = H∖ny, then each element of z is a contrast of the elements of y