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Problem Set 8
Math 423, Section 200, Spring 2015
Due: Monday, April 6.
Review Sections 7B and 7C in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Monday, April 6. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
Recall that if T is an operator on V and λ is an eigenvalue of T , E(λ, T ) = ker(T − λ idV ) is the
λ-eigenspace of T .
1. Let V be a finite dimensional inner product space over C. Show that T ∈ L (V) is normal if
and only if all pairs of eigenvectors corresponding to different eigenvalues are orthogonal and
V = E(λ1 , T ) ⊕ · · · ⊕ E(λm , T ),
where λ1 , . . . , λm are the distinct eigenvalues of T .
2. Let V be a finite dimensional inner product space over R. Show that T ∈ L (V) is self-adjoint
if and only if all pairs of eigenvectors corresponding to different eigenvalues are orthogonal
and
V = E(λ1 , T ) ⊕ · · · ⊕ E(λm , T ),
where λ1 , . . . , λm are the distinct eigenvalues of T .
3. Give an example of a diagonalizable operator T ∈ L (R3 ) that is not self-adjoint with respect
to the Euclidean inner product.
4. Find a ∈ R so that the matrix


1 1 0
0 1 1


1 0 a
represents a normal operator T ∈ L (R3 ) with respect to the standard basis of R3 , where R3 is
considered with the usual Euclidean inner product.
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5.
a. Give an example of a diagonalizable operator T ∈ L (C2 ) (considered with the Euclidean
inner product) whose matrix with respect to the standard basis is symmetric, but which
cannot be diagonalized with respect to an orthonormal basis.
b. Give an example of a non-diagonalizable operator T ∈ L (C2 ) (considered with the
Euclidean inner product) whose matrix with respect to the standard basis is symmetric.
6. Let V be a finite dimensional inner product space, and let T ∈ L (V) be self-adjoint. Suppose
2 and 3 are the only eigenvalues of T . Show that T 2 − 5T + 6 idV = 0.
7. Let V be a finite dimensional inner product space over C. Prove that a normal operator T ∈
L (V) is self-adjoint if and only if all of its eigenvalues are real.
8. Let V be a finite dimensional inner product space over C. Suppose that T ∈ L (V) is a normal
operator such that T 9 = T 8 . Prove that T is self-adjoint and T 2 = T .
9. Let V be a finite dimensional inner product space, T ∈ L (V) self-adjoint, λ ∈ F and ε > 0.
Suppose there exists v such that kvk = 1 and kT v − λvk < ε. Prove that T has an eigenvalue λ0
such that |λ0 − λ| < ε.
10. Let U be a finite dimensional R-vector space, and T ∈ L (U). Prove that T is diagonalizable
if and only if there is an inner product on U that makes T into a self-adjoint operator.
11. Let V be a finite dimensional inner product space over C. Prove that every normal operator on
V has a square root.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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