Page 219 - Advanced Linear Algebra
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Eigenvalues and Eigenvectors 203
((( c ( ( ( ( (( ( (( ( ( (
~ ~
£ £
and thus
( ( c( ( ( ( (8.7 )
~
£
The right-hand side is the sum of the absolute values of all entries in the th row
of except the diagonal entry ( . This sum 9 ² ( ³ is the th deleted absolute
(
(
)
row sum of ( . The inequality 8.7 says that, in the complex plane, the
eigenvalue lies in the disk centered at the diagonal entry ( with radius equal
to 9²(³ . This disk
' c ( ( ( ²(³ ~ ¸' d 9 ²(³¹
GR
is called the Geršgorin row disk for the th row of . The union of all of the
(
Geršgorin row disks is called the Geršgorin row region for .
(
Since there is no way to know in general which is the index for which
(( ( ( , the best we can say in general is that the eigenvalues of ( lie in the
union of all Geršgorin row disks, that is, in the Geršgorin row region of .
(
Similar definitions can be made for columns and since a matrix has the same
eigenvalues as its transpose, we can say that the eigenvalues of lie in the
(
Geršgorin column region of ( . The Geršgorin region . ² ( ³ of a matrix
( 4 ²-³ is the intersection of the Geršgorin row region and the Geršgorin
column region and we can say that all eigenvalues of lie in the Geršgorin
(
region of . In symbols, ( . ( .
(
25. Find and sketch the Geršgorin region and the eigenvalues for the matrix
v y
(~
w z
26. A matrix ( 4 ² ³ is diagonally dominant if for each ~ ÁÃÁ ,
d
( (( 9 ² ( ³
and it is strictly diagonally dominant if strict inequality holds. Prove that
if is strictly diagonally dominant, then it is invertible.
(
d
27. Find a matrix ( 4 ² ³ that is diagonally dominant but not invertible.
d
28. Find a matrix ( 4 ² ³ that is invertible but not strictly diagonally
dominant.
