By Wilfred Kaplan

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Show that if A l . 2, A3 aredistinct, then for each Ak the associated eigenvectors are the vectors cek for c # 0: show that if hi = A? # A3, then thc cigenvectors associatcd with Al are all nonzero vectors r i e l c ~ e 2and thosc associated with A 3 are all nonzero vectors ce3: show that if A1 = A] = A3, then the cigcnvectors associated with A l are all nonzero vectors v = ( v l v l , v3). c ) Let B = diag ( A l , . . , A,). Show that the eigenvectors associatcd with the eigenvalue hi are all nonzero vectors v = ( u l , .

One can introduce coordinates or components on the basis of the axioms themselves. for Rule XI11 guarantees existence of a basis v,, . . , v,,. The Gram-Schmidt process can then be used to construct an orthogonal basis u, . . u,. We "normalize" . 51 52 Advanced Calculus, Fifth Edition these vectors by dividing by their norms to obtain an orthonormal basis: For an arbitrary vector v we can now write, in unique fashion, and call (vr, . . , v;) the components of v with respect to the basis ey, .

A matrix A such that A = A' is called a symmetric matrix. Here A must be a square matrix. The matrix I is symmetric, as are the following matrices: L Also, every diagonal matrix is symmetric. Symmetric matrices are useful in discussing quadratic forms, that is. algebraic expressions of the form For n = 2 the expression is Here ~ 1 x is2 the same as ~ 2 . ~ SO 1 , we could combine the second and third terms. However, it is preferable to split the combined term into two equal terms, each having as coefficient the average of a12 and a2l.

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Advanced Calculus by Wilfred Kaplan


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