Page 250 - 35Linear Algebra

250 Diagonalization

a b
5. When is the 2 × 2 matrix diagonalizable? Include examples in
c d
your answer.
6. Show that similarity of matrices is an equivalence relation. (The defi-
nition of an equivalence relation is given in the background WeBWorK
set.)
7. Jordan form

λ 1
• Can the matrix be diagonalized? Either diagonalize it or
0 λ
explain why this is impossible.
 
λ 1 0
• Can the matrix  0 λ 1  be diagonalized? Either diagonalize
0 0 λ
it or explain why this is impossible.
 
λ 1 0 · · · 0 0
 0 λ 1 · · · 0 0 
 
 0 0 λ · · · 0 0 
• Can the n × n matrix  . . . . . . . . . . be diagonalized?


.
.
 . . . . . . 
 
0 0 0 · · · λ 1
0 0 · · · 0 λ
Either diagonalize it or explain why this is impossible.
Note: It turns out that every matrix is similar to a block ma-
trix whose diagonal blocks look like diagonal matrices or the ones
above and whose off-diagonal blocks are all zero. This is called
the Jordan form of the matrix and a (maximal) block that looks
like
 
λ 1 0 · · · 0
 0 λ 1 0 
 
 . . . . . . 
 . . . 
 
λ
 1 
0 0 0 λ
is called a Jordan n-cell or a Jordan block where n is the size of
the block.

8. Let A and B be commuting matrices (i.e., AB = BA) and suppose
that A has an eigenvector v with eigenvalue λ.

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