Math 251 - Alexiades
Review on material after Exam2
(diagonalization and inner product spaces)
Concepts
  • eigenvalue-eigenvector, eigenspace, algebraic,geometric multiplicities
  • diagonable matrix, diagonalization
  • inner product axioms ; norm and distance from an inner product
  • orthogonal, orthonormal set ; orthogonal complement of a subspace, projection
  • change of basis
  • orthogonal matrix

    Precise definition of
  • eigenspace Eλ:   the subspace (of Rn) spanned by the eigenvectors of an eigenvalue λ.
  • algebraic multiplicity of eigenvalue λ:  multiplicity of λ as a root of the characteristic polynomial.
  • geometric multiplicity of eigenvalue λ:   dimension of the eigenspace Eλ of λ.
  • diagonable n×n matrix: ∃ invertible P such than P−1A P = Λ=diagonal   (A similar to diagonal matrix Λ)
  • inner product   (3 axioms)
  • orthogonal set {v1,v2,...,vk} :   <vi,vj > = 0 for i≠j.
  • orthonormal set {v1,v2,...,vk} :   <vi,vj > = δij  =0 for i≠j , =1 for i=j.
  • orthogonal complement of a subspace W:   set of all vectors in V that are orthogonal to every vector in W.
  • orthogonal n×n matrix:   A−1 = AT   (equivalently:   ATA = In).

    Methods: Know How To
  • decide if a matrix is diagonable, and diagonalize it (find the P and Λ)
  • decide if something is a norm, an inner product
  • decide if a set {v1,v2,...,vk} is orthogonal, orthonormal
  • construct an orthonormal set from a lin.independent set (Gram-Schmidt process)
  • find coordinates w.r.t. an orthonormal basis
  • find projection onto a subspace (via an ON basis)
  • find projection onto a column space (via Least Squares)
  • decide if a matrix is orthogonal

    Basic / crucial Theorems to know
  • 5.1.3 , 5.2.1 , 5.2.2
  • Cauchy-Schwarz(6.2.1), Pythagorean Thm(6.2.3:holds iff vectors are orthogonal)
  • 6.3.1 , 6.3.2 , 6.3.3 , 6.3.4
  • 6.4.1, 6.4.2