Math 4153 Exam 3 Review
The syllabus for Exam 3 is Chapter 6 (pages 110-21), Chapter 7 through page 137, and
Chapter 8 through page 182 in Axler.
1. You should be sure to know precise definition of the terms we have used, and you should
know precise statements (including all relevant hypotheses) for the main theorems
proved. Know how to do all of the homework problems.
2. Outline of subjects for Exam 3:
∙ Eigenvalues, eigenvectors, and eigenspaces
∙ Inner product spaces, norms
∙ Orthogonal complements and projections
∙ Adjoint operators and matrices in orthonormal bases
∙ Self-adjoint and normal operators
∙ The spectral theorem for self-adjoint and normal operators on complex vector
spaces
∙ The spectral theorem for self-adjoint operators on real vector spaces
∙ Generalized eigenvectors and generalized eigenspaces
∙ The fact that the generalized eigenspace 𝑉 (𝜆) = (𝑇 − 𝜆𝐼)dim 𝑉
∙ Nilpotent operator
∙ The decomposition theorem (Theorem 8.23, Page 174).
∙ Multiplicity of an eigenvalue (Page 171)
∙ Characteristic polynomial of an operator 𝑇 .
∙ Cayley-Hamilton Theorem
∙ Block decomposition of an operator into blocks which are upper triangular (Theorem 8.28)
∙ Square root of operators (Theorem 8.32)
∙ Minimal polynomial of an operator 𝑇
3. Definitions (there is a lot of repetition with the above section):
∙ Eigenvalues and eigenvectors of linear transformations and matrices
∙ Generalized eigenvectors and eigenspaces of linear transformations and matrices
∙ Invariant subspace 𝑈 ⊆ 𝑉 under a linear operator 𝑇 ∈ ℒ(𝑉 )
∙ Inner Product and inner product space
∙ Norm associated to an inner product space
∙ Orthogonal and orthonormal lists and bases
∙ Orthogonal complement
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Math 4153 Exam 3 Review
∙ Projection and orthogonal projection
∙ Adjoint operator
∙ Self-adjoint and normal operator
∙ Characteristic polynomial of a complex operator or matrix in terms of uppertriangular form
∙ Nilpotent operators and matrices (𝑇 is nilpotent means 𝑇 𝑚 = 0 for some 𝑚 ≥ 1)
∙ Characteristic polynomial of an upper-triangular matrix or of a linear
transfor⊕
mation with an ∏
upper triangular matrix form. That is, if 𝑉 =
𝜆 𝑉 (𝜆), then
dim 𝑉 (𝜆)
𝜒𝑇 (𝑥) = 𝑞(𝑧) = 𝜆 (𝑧 − 𝜆)
.
∙ Minimal polynomial (Page 179)
4. Computations: know how to compute
∙ Given an eigenvalue, compute the eigenspace using Gaussian elimination.
∙ Compute the orthogonal complement of a vector or a vector subspace
∙ Compute the orthogonal projection of a vector to a subspace
∙ Given the eigenvalues, compute the generalized eigenspaces using Gaussian elimination
∙ Compute the characteristic polynomial of a linear transformation or matrix
5. Major results. Know the statements, but don’t memorize them word for word–know
what they say mathematically. Especially important ones are in bold.
∙ Orthogonal complements: Theorem 6.29
∙ Existence of Adjoints: Theorem 6.45 and following discussion.
∙ The spectral theorem: Theorems 7.9 and 7.13.
∙ The decomposition theorem: Theorem 8.23.
∙ Statement that dim(𝑉 (𝜆)) = 𝑑𝜆 : Theorem 8.10.
∙ Square roots: Theorem 8.32 and Exercise 11, page 159.
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Math 4153 Exam 3 Review
Review Exercises
1. True/False: If 𝑇 is normal on a finite-dimensional complex inner product space, then
all generalized eigenvectors of 𝑇 are eigenvectors.
2. True/False: If (𝑣1 , 𝑣2 ) is an arbitrary basis of ℂ2 with the standard inner product, then
𝑡
ℳ(𝑇 ∗ , (𝑣1 , 𝑣2 )) = ℳ(𝑇, (𝑣1 , 𝑣2 )) .
3. True/False: If 𝑇 is self-adjoint, then 𝑇 is normal.
4. True/False: If 𝑇 is normal, then 𝑇 is self-adjoint.
5. True/False: If a linear operator 𝑇 has a nonzero generalized eigenvector of eigenvalue
𝜆, then it also has a nonzero eigenvector of eigenvalue 𝜆.
6. If 𝑉 is a finite dimensional complex vector space and 𝑇 ∈ ℒ(𝑉 ), prove that 𝑇 is
nilpotent if and only if the characteristic polynomial of 𝑇 is 𝑧 𝑚 where 𝑚 = dim 𝑉 .
7. Let 𝑉 be a finite-dimensional inner product space, and let 𝑉 = 𝑈 ⊕ 𝑈 ⊥ . Show that
the orthogonal projection operator 𝑃𝑈, 𝑈 ⊥ is self-adjoint.
8. For the matrix
⎡
1
⎢0
𝐴=⎢
⎣0
0
⎤
2 2 4
1 −2 6⎥
⎥
0 1 3⎦
0 0 0
compute the eigenvalues, the characteristic polynomial, the multiplicity of each eigenvalue, and bases for each generalized eigenspace.
9. The matrix
⎡
⎤
2 −1 0 1
⎢0 3 −1 0⎥
⎥
𝐴=⎢
⎣0 1
1 0⎦
0 −1 0 3
has characteristic polynomial 𝑞(𝑧) = (𝑧 − 2)3 (𝑧 − 3). Determine the eigenvalues, the
multiplicity of each eigenvalue, and bases for each generalized eigenspace.
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