2Orthogonal Matrices and Change of Basis Let be an ordered basis for the vector space . Recall that the coordinate matrix of a vector in is the column vector
If is another basis for , then the transition matrix from to changes a coordinate matrix relative to into a coordinate matrix relative to ,
The question you will explore now is whether there are transition matrices that preserve the length of the coordinate matrix—that is, given , does ?
For example, consider the transition matrix from Example 5 in Section 4.7,
relative to the bases for ,
and
If , then and . (Verify this.) So, using the Euclidean norm for ,
You will see in this project that if the transition matrix is orthogonal, then the norm of the coordinate vector will remain unchanged. You may recall working with orthogonal matrices in Section 3.3 (Exercises 73, 74, 75, 76, 77, 78, 79, 80, 81, and 82) and Section 5.3 (Exercise 65).
Definition of Orthogonal Matrix The square matrix is orthogonal when it is invertible and .
Show that the matrix defined previously is not orthogonal.
Show that for any real number , the matrix
is orthogonal.
Show that a matrix is orthogonal if and only if its columns are pairwise orthonormal.
Prove that the inverse of an orthogonal matrix is orthogonal.
Is the sum of orthogonal matrices orthogonal? Is the product of orthogonal matrices orthogonal? Illustrate your answers with appropriate examples.
Prove that if is an orthogonal matrix, then for all vectors in .
Verify the result of part 6 using the bases and .
See image: Orthogonal Matrices and Change of Basis
