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Carlo Beenakker
Carlo Beenakker
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see the appendix of this paper for understanding haarHaar measure.."Determinantal point processes in the plane from products of random matrices" : http://www.imstat.org/aihp/accepted.htmlDeterminantal point processes in the plane from products of random matrices

intuition for haarHaar random orthogonal matrix: choose a vector randomly from the unit sphere in R^n${\mathbb R}^n$ ( uniformuniform distribution on the unit sphere). Thats urThat's the first column. Now for the second column, choose a vector randomly from the unit sphere in n-1the $n-1$ dimensional subspace orthogonal to the first column. similarlySimilarly for the third column, choose a vector randomly from the unit sphere in n-2the $n-2$ dimensional subspace orthogonal to the first two columns...and so on....

see the appendix of this paper for understanding haar measure.."Determinantal point processes in the plane from products of random matrices" http://www.imstat.org/aihp/accepted.html

intuition for haar random orthogonal matrix: choose a vector randomly from unit sphere in R^n( uniform distribution on unit sphere). Thats ur first column. Now for second column, choose a vector randomly from unit sphere in n-1 dimensional subspace orthogonal to first column. similarly for third column, choose a vector randomly from unit sphere in n-2 dimensional subspace orthogonal to first two columns...and so on....

see the appendix of this paper for understanding Haar measure: Determinantal point processes in the plane from products of random matrices

intuition for Haar random orthogonal matrix: choose a vector randomly from the unit sphere in ${\mathbb R}^n$ (uniform distribution on the unit sphere). That's the first column. Now for the second column, choose a vector randomly from the unit sphere in the $n-1$ dimensional subspace orthogonal to the first column. Similarly for the third column, choose a vector randomly from the unit sphere in the $n-2$ dimensional subspace orthogonal to the first two columns...and so on....

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Nanda Kishore
Nanda Kishore
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see the appendix of this paper for understanding haar measure.."Determinantal point processes in the plane from products of random matrices" http://www.imstat.org/aihp/accepted.html

intuition for haar random orthogonal matrix: choose a vector randomly from unit sphere in R^n( uniform distribution on unit sphere). Thats ur first column. Now for second column, choose a vector randomly from unit sphere in n-1 dimensional subspace orthogonal to first column. similarly for third column, choose a vector randomly from unit sphere in n-2 dimensional subspace orthogonal to first two columns...and so on....