Orthonormality means that if and if These equations are equivalent to orthogonality ofĮxercise 4. Conversely, rows or columns of an orthogonal matrix form an orthonormal basis. If is an orthonormal basis, then the matrix is orthogonal. Orthogonal matrix is a real square matrix whose product, with its transpose, gives an identity matrix. Strictly speaking, in case we have rotation and in case - rotation combined with reflection.Īnother interpretation is suggested by the next exercise.Įxercise 3. The determinant of an orthogonal matrix is equal to 1 or -1. Since the origin is unchanged under any linear mapping, Exercise 2 gives the following geometric interpretation of an orthogonal matrix: it is rotation around the origin (angles and vector lengths are preserved, while the origin stays in place).
For any vectors scalar products are preserved: Therefore vector lengths are preserved: Cosines of angles are preserved too, because Thus angles are preserved. This means it has the following features: it is a square matrix. An orthogonal matrix preserves scalar products, norms and angles. In an orthogonal matrix, the columns and rows are vectors that form an orthonormal basis. d) Just apply to the definition to getĮxercise 2. Such matrices are usually denoted by the letter Q. a) is the left inverse of Hence, is invertible and its inverse is b) from the inverse definition. Definition: if the columns of a matrix are orthonormal, the matrix itself is called orthogonal. Then a) b) the transpose is orthogonal, c) the inverse is orthogonal, d) A square matrix is called orthogonal ifĮxercise 1.
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