# Orthogonal Matrix

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## orthogonal matrix

[ȯr′thäg·ən·əl ′mā·triks] (mathematics)

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 *A*′ *A* = *E*. The elements of an orthogonal matrix satisfy the relations

or the equivalent relations

The determinant ǀ*A*ǀ 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 *a _{ij}* in the coordinate transformation equations

form an orthogonal matrix.