Diagonalization Theorems (2pp.)

hogheavyweightElectronics - Devices

Oct 8, 2013 (3 years and 8 months ago)

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Diagonalization TheoremsTheorem 3(Diagonalization Theorem)(a)AnmmmatrixAisdiagonableif and only ifAhasmlinearly independent eigenvectors.ofA.(b)Suppose~v
1
;:::;~v
m
2 C
mis alinearly inde-
pendent set ofeigenvectorsofAwith corresponding
eigenvalues 
1
;:::;
m(soA~v
i
= 
i
~v
i,for1  i 
m ).Then the matrixP = (~v
1
j:::j~v
m
)isinvertibleand we haveP
1
AP = Diag(
1
;:::;
m
):Remark:Writech
A
(t) = (t 
1
)
m
1
   (t 
r
)
m
r
;where the
i's aredistinct,and letB
ibe abasisof
the corresponding eigenspaces E
A
(
i
),1  i  r.
Then it can be shown that B = B
1
[B
2
[:::[B
ris alinearly independentset.Thus:Aisdiagonable,Bis abasisofC
m,#B = m,dimE
A
(
1
) +:::+dimE
A
(
r
) = m:1
Corollary:IfAhasmdistincteigenvalues (i.e.ifch
A
(t)hasmdistinctroots),thenAisdiagonable.Warning:Theconverseof this corollary isfalse:
a matrix Acan bediagonableyethaverepeatedeigenvalues.Example:them  midentity matrixIis diago-
nal (hence diagonable),but has onlyoneeigenvalue
1
= 1(repeatedmtimes).Remark:IfAis arealmatrix (i.e.all entries ofAare
real numbers) and is symmetric,i.e.A
t
= A,thenAisautomatically diagonable,as the the following
useful result shows:Theorem 4(Principal Axis Theorem)IfAis areal symmetricmatrix,thenAisorthog-
onally diagonable;in other words,there exists anorthogonalmatrixP(i.e.a real matrix satisfyingP
1
= P
t) such thatP
1
APis a diagonal matrix.Remark:Thenameof this theorem comes from the
fact that this theoremcan used to showthatquadricsinR
ncentered at the origin (e.g.ellipsesinR
2,el-
lipsoids inR
3,etc.) can berotatedso that theirprincipal axesare along thecoordinate axesofR
n.2