## Transcribed Text

Question 1 Find all square roots of the matrix
5 2
.
3 0
That is to say find all matrices B such that B² - A. Hint: Finding the
square root of a diagonal matrix is easy. You can leave the answer as a
product.
Question 2 Use the Cauchy-Schwarz inequality to prove that for 0 < a < 1 we have
that
/2
- 1
J0
Question 1 True or false:
(a) Every linear operator in an n-dimensional vector space has n distinct
eigenvalues.
(b) There exists a square real matrix with no real eigenvalues.
(c) There exists a square matrix with no (complex) eigenvalues.
(d) Similar matrices always have the same eigenvalues.
(e) Similar matrices always have the same eigenvectors.
(f) The sum of two eigenvectors of a matrix is A is also an eigenvector of
A.
Question 2 Find the characteristic polynomials, eigenvalues and eigenvectors of the
following matrices:
1 3 3
(==)
4
2
3
,
( - 2 1 4 1
3 - 5 -3
,
.
3 3 1
Question 3 Recall that a linear operator A is nilpotent if Ak: = 0 for some integer k > 1.
Prove that if A is nilpotent, then o(A)=(0). =
Question 4
Let A be an n X n matrix. True or false:
(a) AT has the same eigenvalues as A.
(b) AT has the same eigenvectors as A.
(c) If A is diagonalisable, then so is AT.
Question 5 Suppose that A is a square real matrix. Show that if \ € C \ R is an
eigenvalue of A with eigenvector U € Cn, then I is an eigenvalue of A with
eigenvector U.
Question 6 Construct a matrix A with eigenvalues 1 and 3 with corresponding eigen-
vectors (1,2) and (1,1), respectively. Is this matrix unique?
Question 7 Diagonalise the following matrices if possible:
4 1 - 1 2 ). , 6 4 1
.
Question 8 Consider the matrix
2 6 -6
0 5 -2
0 0 4
(a) Find the eigenvalues of A. Is it possible to find them without comput-
ing?
(b) Argue, without computing anything, that A is diagonalisable.
(c) Diagonalise A.
Question 9 Find all square roots of the matrix
5 2
.
-3 0
That is to say find all matrices B such that B² - A. Hint: Finding the
square root of a diagonal matrix is easy. You can leave the answer as a
product.

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