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Special Orthogonal Matrix

A square matrix A is a special orthogonal matrix if

AA^(T)=I,
(1)

where I is the identity matrix, and the determinant satisfies

detA=1.
(2)

The first condition means that A is an orthogonal matrix, and the second restricts the determinant to +1 (while a general orthogonal matrix may have determinant -1 or +1). For example,

1/(sqrt(2))[1 -1; 1 1]
(3)

is a special orthogonal matrix since

[1/(sqrt(2)) -1/(sqrt(2)); 1/(sqrt(2)) 1/(sqrt(2))][1/(sqrt(2)) 1/(sqrt(2)); -1/(sqrt(2)) 1/(sqrt(2))]=[1 0; 0 1]
(4)

and its determinant is 1/2-(-1/2)=1. A matrix m can be tested to see if it is a special orthogonal matrix using the Wolfram Language code 

  SpecialOrthogonalQ[m_List?MatrixQ] :=
    (Transpose[m] . m == IdentityMatrix @ Length @ m
&& Det[m] == 1)

The special orthogonal matrices are closed under multiplication and the inverse operation, and therefore form a matrix group called the special orthogonal group SO(n).

See also

Antisymmetric Matrix, Inner Product, Orthogonal Group, Orthogonal Matrix, Orthogonal Transformation, Special Linear Matrix, Special Orthogonal Group, Spin Group, Unitary Matrix

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Special Orthogonal Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SpecialOrthogonalMatrix.html

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