# Write a program (use R, Matlab, Python, Fortran, C) to generate n×n random orthogonal matrices with.

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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