Orthogonal Matrix

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orthogonal matrix

[ȯr′thäg·ən·əl ′mā·triks]
A matrix whose inverse and transpose are identical.

Orthogonal Matrix


An orthogonal matrix of order n is a matrix

whose product with the transpose A′ gives the identity matrix, that is, AA′ = E and AA = E. The elements of an orthogonal matrix satisfy the relations

or the equivalent relations

The determinant ǀAǀ of an orthogonal matrix is equal to +1 or – 1. The product of two orthogonal matrices is an orthogonal matrix. All orthogonal matrices of order n form, with respect to the operation of multiplication, a group called the orthogonal group. In conversion from one rectangular coordinate system to another, the coefficients aij in the coordinate transformation equations

form an orthogonal matrix.

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