The rest of the matrix is an orthogonal matrix; thus O ( " n " ) is a subgroup of ( and of all higher groups ).
42.
More explicitly : For every symmetric real matrix " A " there exists a real orthogonal matrix " Q " such that is a diagonal matrix.
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Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition.
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Similarly, SO ( " n " ) is a subgroup of; and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure.
45.
Thus, negating one column if necessary, and noting that a reflection diagonalizes to a + 1 and " 1, any orthogonal matrix can be brought to the form
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:Alternatively we multiply by an orthogonal matrix with determinant & minus; 1 ( corresponding to a reflection in a line through the origin ), followed by a translation.
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With an orthogonal matrix substituted into the multivariate standard normal formula things cancel and I get a numerator that works out to be multivariate standard normal which is what I want.
48.
A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group; see Rotation matrix # Uniform random rotation matrices.
49.
The orthogonal matrices whose determinant is + 1 form a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns.
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Note that for any orthogonal matrix Q, if we set L = LQ and F = Q ^ T F, the criteria for being factors and factor loadings still hold.
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