Math 369 HW #7
Due at the beginning of class on Friday, October 24
Reading: Sections EE, PEE.
Problems:
We say that a square matrix M is similar to the square matrix N if there is an invertible matrix
A so that M = AN A−1 . Other terms for this: we say that N is a “change of basis” of M or that
N is “conjugate” to M .
a 0
1. Suppose M =
(such a matrix is sometimes called a “scalar matrix”). Are there any
0 a
other matrices (i.e., not equal to M ), which are similar to M ? Find one if “yes”, and prove
that there can’t be any if “no”.
a1 0
. Are there any other
2. Suppose M is a 2 × 2 diagonal matrix, meaning that M =
0 a2
matrices which are similar to M ? Find one if “yes”, and prove that there are none if “no”.
3. Suppose M is similar to N . Is N similar to M ? Prove or give a counterexample.
4. Let f : R2 → R2 be the linear transformation which rotates objects in the plane around the
origin by 30 degrees counterclockwise.
0
1
.
,
(a) Give a matrix F for f with respect to the standard basis
1
0
( √ ! )
3
0
2
(b) Give a matrix G for f with respect to the basis
.
,
1
1
2
(c) Since the matrices in (a) and (b) are two different matrices for the same linear transformation, they must be similar. So find an invertible matrix A so that G = AF A−1 .
5. For each of the following, find the requested matrix and draw a picture showing what the
transformation does.
*
*
*
*
(a) Find a 2 × 2 matrix A for a linear transformation R having the property that R( u) = λ u
*
*
for some constant λ only if u = 0 . (Think of rotations)
(b) Find a 2 × 2 matrix
B for a linear transformation S having the property that S( u) = λ u
a
*
only when u =
for some a ∈ R. (Think of sheering maps)
0
1