Write a

program (use R, Matlab, Python, Fortran, C) to generate n×n random orthogonal

matrices with the Haar uniform distribution. Use the following method due to

Heiberger (1978), which was modified by Stewart (1980). (See also Tanner and

Thisted 1982.)

b) Let

ri = xi2, and let Hi be the i × i reflection matrix that transforms xi into the

i-vector (ri, 0, 0,…, 0).

c) Let

Hi be the n × n matrix

d)

Deliver the orthogonal matrix Q = JH1H2

»

Write a

program (use R, Matlab, Python, Fortran, C) to generate n×n random orthogonal

matrices with the Haar uniform distribution. Use the following method due to

Heiberger (1978), which was modified by Stewart (1980). (See also Tanner and

Thisted 1982.)

b) Let

ri = xi2, and let Hi be the i × i reflection matrix that transforms xi into the

i-vector (ri, 0, 0,…, 0).

c) Let

Hi be the n × n matrix

d)

Deliver the orthogonal matrix Q = JH1H2 ··· Hn.

The

matrix Q generated in this way is orthogonal and has a Haar distribution.

Can you

think of any way to test the goodness-of-fit of samples from this

algorithm?

Generate a sample of 1,000 2×2 random orthogonal matrices,

and

assess how well the sample follows a Haar uniform distribution.

»

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