MATH 436 Pennsylvania State University Linear Done Right Linear Algebra Question

There are five questions. The questions are the content in the chapter 8 and 10 in the book “Linear done right”. Please answer it as specify as possible

1. Define a linear operator T on R3 such that T is nilpotent and satisfies
(C) = 0
T
(8) – 3
2. Find all possible Jordan canonical forms of 4 x 4 complex matrices which satisfy the
following three conditions simultaneously:
• A is not diagonalizable;
• A has characteristic polynomial (1 – 5)²(x – 1)?;
• A satisfies the equation (A – 51)(A – 1) = 0.
3. Let V = Rº be the inner product space equipped with the inner product
12
Y2
2:01:22 +242 +2241 + yıy2.
Let T be the linear operator defined by
T
(
©) = (2+2)
Answer the following two questions with justifications.
(a) Is T diagonalizable?
(b) Does there exist an orthonormal basis of V consisting of eigenvectors of T? (Or-
thonormal with respect to the above inner product.)
4. Let V be the vector space of nxn complex matrices. Let A be a fixed matrix. Define
a map T :V + V by T(B) = AB – BA.
(a) Verify that T is a linear operator on V.
(b) Prove that if A is a nilpotent matrix, then T is a nilpotent operator.
0
5. (a) Let A=
be a diagonal complex matrix and f(2) be a polynomial
0 An
with coefficients in C. Describe f(A).
(b) Let T be a normal operator on an n-dimensional complex inner product space.
Prove that there is a polynomial f, with complex coefficients, such that
f(T) = T”
(You may use without proof the following fact: Given any two n-tuples of complex
numbers (41, …, In) and (H1, …,Hr) there is a polynomial f(1), with complex
coefficients, such that f(1) = M1, …, f (An) = for.)