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Unformatted text preview: 5 Change of Basis In many applications, we may need to switch between two or more different bases for a vector space. So it would be helpful to have formulas for converting the components of a vector with respect to one basis into the corresponding components of the vector (or matrix of the operator) with respect to the other basis. The theory and tools for quickly determining these change of basis formulas will be developed in these notes. 5.1 Unitary and Orthogonal Matrices Definitions Unitary and orthogonal matrices will naturally arise in change of basis formulas. They are defined as follows: A is a unitary matrix A is an invertible matrix with A 1 = A , and A is an orthogonal matrix A is an invertible real matrix with A 1 = A T A is a real unitary matrix . Because an orthogonal matrix is simply a unitary matrix with real-valued entries, we will mainly consider unitary matrices (keeping in mind that anything derived for unitary matrices will also hold for orthogonal matrices after replacing A with A T ). The basic test for determining if a square matrix is unitary is to simply compute A and see if it is the inverse of A ; that is, see if AA = I . ! Example 5.1: Let A = 3 5 4 5 i 4 5 i 3 5 Then A = 3 5 4 5 i 4 5 i 3 5 9/21/2011 Change of Basis Chapter &amp; Page: 52 and AA = 3 5 4 5 i 4 5 i 3 5 3 5 4 5 i 4 5 i 3 5 = = I . So A is unitary. Obviously, for a matrix to be unitary, it must be square. It should also be fairly clear that, if A is a unitary (or orthogonal) matrix, then so are A , A T , A and A 1 . ? Exercise 5.1: Prove that, if A is a unitary (or orthogonal) matrix, then so are A , A T , A and A 1 . The term unitary comes from the value of the determinant. To see this, first observe that, if A is unitary then I = AA 1 = AA . Using this and already discussed properties of determinants, we have 1 = det ( I ) = det ( AA ) = det ( A ) det ( A ) = det ( A ) det ( A ) = | det ( A ) | 2 . Thus, A is unitary equal1 | det A | = 1 . And since there are only two real numbers which have magnitude 1 , it immediately follows that A is orthogonal equal1 det A = 1 . An immediate consequence of this is that if the absolute value of the determinant of a matrix is not 1 , then that matrix cannot be unitary. Why the term orthogonal is appropriate will become obvious later. Rows and Columns of Unitary Matrices Let U = u 11 u 12 u 13 u 1 N u 21 u 22 u 23 u 2 N . . . . . . . . . ....
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