# Download PDF by Morris L. Eaton: Multivariate Statistics: A Vector Space Approach

By Morris L. Eaton

ISBN-10: 0940600692

ISBN-13: 9780940600690

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X , ) for V and real numbers A,, . . , A, such that Further, A , , . . , A , are the eigenvalues of A and Ax, = A,x,, i = 1,. . , n. Proof: The proof of the first assertion is by induction on dimension. For n = 1, the result is obvious. Assume the result is true for integers 1,2,. . , n - 1 and consider A E C(V, V), which is self-adjoint on the inner product space (V, (. )), n = dim V. Let A be an eigenvalue of A . 41, there exists u E V, v * 0, such that A v = Av. Set x , = v/llvll and A, = A.

X,) would simply be en. a, where the elements not indicated are zero. Of course, this says that the linear transformation is A, times the identity transformation when restricted to span{x,). Unfortunately, it is not possible to find such a basis for each linear transformation. However, the numbers A,, . . , A,, which are called eigenvalues after we have an appropriate definition, can be interpreted in another way. Given A E a , Ax = Ax for some nonzero vector x iff (A A I ) x = 0, and this is equivalent to saying that A - A I is a singular matrix, 44 VECTOR SPACE THEORY that is, det(A - AI) x + 0 such that Ax = = calculation shows that 0.

If D is n-linear and satisfies (ii), D is sometimes called an alternating n-linear function, since D(A) changes sign if two columns of A are interchanged. The basic result that relates all determinant functions is the following. 32. The set of determinant functions is a one-dimensional complex vector space. If D is a determinant function and D s 0, then D ( I ) * 0 where I is the n X n identity matrix in e n . Proof. We briefly outline the proof of this proposition since the proof is instructive and yields the classical formula defining the determinant of an n x n matrix.