Linear algebra/Orthogonal matrix

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A real-valued and square matrix is orthogonal (or orthonormal) if where is the transpose and is its inverse. Equivalently, is orthogonal when where is the identity matrix. It has a number of well-known properties:[1]

Example[edit | edit source]

Since orthogonal matrices form a group under multiplication, we can construct a non-trivial orthogonal matrix my multiplying two matrices that are easy to understand. Consider, for example,

Both and posses a symmetry that could lead one to postulate a non-existent symmetry among off-diagonal elements. But no such symmetry exists for . The upper-left 2x2 submatrix in represent a 45 degree rotation around the (or ) axis, plus and inversion through that same axis, i.e., .

Notes[edit | edit source]