By Wilfred Kaplan
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Extra info for Advanced Calculus
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.
Advanced Calculus by Wilfred Kaplan
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