Some Conditions Equivalent to Diagonalizability
of an 8 ‚ 8 Matrix E
E is diagonalizable
Ì Ë
There is a basis of eigenvectors of E
for ‘8
Ì Ë
E has 8 linearly independent
eigenvectors
Ì Ë
The sum of the dimensions of the eigenspaces
of E œ 8
Suppose E is % ‚ %
Can you say whether or not E is diagonalizable from
the information given?
1. The characteristic polynomial of E is
GÐ-Ñ œ Ð-  "ÑÐ-  #ÑÐ-#  -  "Ñ
2. E has eigenvalues - œ  #ß $ß " and 4
3. E has eigenvalues - œ  #ß $ß " and these have
multiplicities "ß #ß " in that order.
_________________________________________
(For after today's lecture)
"
!
4. E is similar to 
!
!
#
#
!
!
%
"
&
!
 &
$

"
'
5. What does diagonalizable mean for a " ‚ "
matrix. Think it through just to see that all the pieces
of the more general theory fit together correctly.
X
Ò
Example
Z
Basis
Æ Ò ÓU
[
Æ Ò ÓV
U œ Ö , " ß , # ß ,$ ×
‘$
Suppose
Basis
V œ Ö-" ß - #ß -$ ß -% ×
‘%
Ò
X Ð," Ñ œ -#  %-$
X Ð,# Ñ œ -"  $-#  %-$  -%
X Ð,$ Ñ œ &-"  #-$  #-%
Then Q œ Ò ÒX Ð," ÑÓV ÒX Ð," ÑÓV ÒX Ð,$ ÑÓV Ó


œ
!
"
%
 !
"
$
%
"
&
!

#
 #
So if @ œ #,"  (,#  ',$
then


ÒX Ð@ÑÓV œ 
!
"
%
 !
"
$
%
"
&
  $( 
#
   "* 
! 
 ( œ

# 
$#
 ' 

#
&
so X Ð@Ñ œ  $(-"  "*-#  $#-$  &-%
#
X ÀZ œ‘
Example Suppose
Ä
 B 
B
[ œ ‘ is given by X   œ B  C .
C
B  C
$
"
The standard matrix for X is E œ Ò X Ð/" Ñ X Ð/# Ñ Ó œ "
"
! 
"
 "
Choose two new bases:
"
"
U œ Ö," ß ,# × œ Ö ß 
×
#
" 
For ‘#
! !  " 
V œ Ö-" ß -# ß -$ × œ Ö ! ß # ß  " ×
" "  ! 
$
For ‘
With respect to the bases U , V what is the matrix for X À Q œ ÒX ÓUßV œ ??
Z œ ‘#
X œ mult by E
Ä
Æ Ò ÓU
‘#
[ œ ‘$
Æ Ò ÓV
mult by Q œ ??
Ä
‘$
Q œ Ò ÒX Ð," ÑÓV ÒX Ð,# ÑÓV Ó
(OVER)
 -" 
To find ÒX Ð," ÑÓV œ -# ß we need to find the weights -" ß -# ß -$ so that
 -# 
 " 
!
!
 " 
"
$
X Ð ," Ñ œ X Ð   Ñ œ
œ -" !  -# #  -$  " :
#
  "
"
"
 ! 
!
so we need to solve !
"
!
#
"
"  -"   " 
"
-# œ
$ Þ



!
-$
 "
 ." 
Similarly to find ÒX Ð,# ÑÓV œ .# we need to find the weights ." ß .# ß .$ so that
 .# 
  "
!
!
 " 
"
!
!
#
" :
X Ð,# Ñ œ X Ð
Ñ
œ
œ
.

.

.
"
#
$
" 
  #
"
"
 ! 
!
so we need to solve !
"
!
#
"
"  ."    " 
"
.# œ
! Þ
!  .$    # 
We solve both systems simultaneously by row reducing the “double-augmented” matrix:
!
!
"
!
#
"
"
"
!
"
$
"
 "
"
!
µ ÞÞÞ µ  !
 #
!
!
"
!
!
!
"
$
#
"
 $# 
 "# 
"
$
 #
 -" 
  $
 ." 
#
Therefore -# œ ÒX Ð," ÑÓV œ
and .# œ ÒX Ð,# ÑÓV œ   "# Þ
 -$ 
 " 
 .$ 
 "
 $
So the matrix for X with respect to bases U and V is À Q œ  #
 "
 $# 
 "# Þ
"