Chapter 5 Similar Diagonal Matrix(continued)
1.Theorem Let A be an n n ? matrix and Λbe a diagonal matrix ，then
Λ~A (A is diagonalizable 可对角化)?A has n linearly independent eigenvectors.
Example1 Suppose
.320230221????
? ??--=A
(1)Show that A is diagonalizable. (2)Find C and Λ such that .1Λ=-AC C (3)Find .k A (k is a positive integer.)
Solution (1)A I -λ3
2
23
02
21
-----=
λλλ)5()1(2--=λλ0= .5,1321===∴λλλ
For ,121==λλthe corresponding homogeneous system is
O X A I =-)(
that is,
O x x x =????
? ??????? ??----32122022022
0 ∴ A system of fundamental solutions is
????
?
??-=????? ??=110,00121X X
For ,53=λ the corresponding homogeneous system is
O X A I =-)5(
that is,
O x x x =????
? ??????? ??--32122022022
4
????? ??-→-0002202245A I ????? ??-→000220404???
?? ?
?-→000110101
∴ A system of fundamental solutions is
????
?
??-=1113X
(2) Note (3)
Example2 Let
????
? ??---=322232221A
is A similar to a diagonal matrix?
Solution A I -λ3
2
2
23
22
21
------=
λλλ)5()1(2--=λλ0= .5,1321===∴λλλ
For ,121==λλthe corresponding homogeneous system is
O X A I =-)(
that is,
O x x x =???
?
? ??????? ??-----321222222220
????? ??-----→-222220222A I ???
??
??----→440220222
????? ??--→000220222????? ??→000220002????
?
??→000110001 ∴ A system of fundamental solutions is
????
?
??-=1101X
For ,53=λ the corresponding homogeneous system is
O X A I =-)5(
that is,
O x x x =????
? ??????? ??---321222222224
????? ??-→-0002222245A I ????? ??-→000310224????? ??-→000310804???
?
? ??-→000310201
∴ A system of fundamental solutions is
????
? ??-=1322X
2.Theorem If A is n n ? and A has n distinct eigenvalues, then A is diagonalizable.
(反之不成立) 3. Definition Suppose ()().,,,,,,,2121n T
n T
n R b b b a a a ∈== βαWe define an
inner product(内积) by .),(2211n n b a b a b a +++= βα Note 1) If both βαand are row vectors, then .),(T αββα=
2)If both βαand are column vectors, then .),(βαβαT = 4. Properties of Inner Products
()()αββα,,)1=
).,(),(),()2βαβαβαk k k ==
3) If (),,,,21n c c c =γthen
()()γβγαγβα,,),(+=+ and ()()βγαγβαγ,,),(+=+
0),()4≥αα
0),(0=?=ααα
0),(0>?≠ααα Proof 1) 2) 3) 4)
5. Definition The length of α is given by ().,a a =
α
Note 1) α is called an identity vector(unit vector) if .1=α 2) ()(),,,2ααk a a k ka ka k ===
where k is a number.
3)For any nonzero vector α
αααααα?==11,
is an identity vector. 4) βαand are said to be orthogonal(正交) if (),0,=βαdenoted by .βα⊥ If ,0=αthen αis orthogonal to any vector.
6. Definition 1)A vector set is called an orthogonal vector set if it consists of
pairwise orthogonal nonzero vectors. (由非零的两两正交的向量组组成的向量组称为正交向量组)
2)An orthonormal set of vectors(标准正交向量组) is an orthogonal set of unit vectors.
If s ααα,,,21 is an orthonormal set of vectors, then
()),,2,1,(01,s j i j
i j
i j i =?
??≠==αα
Example
is an orthonormal set of vectors.
7.Theorem If r ααα,,,21 are pairwise orthogonal nonzero vectors, then
r ααα,,,21 are linearly independent.
Note The vectors of an orthogonal vector set are linearly independent. Proof
Exercises
1.Is A similar to a diagonal matrix? If so, find C and Λ such that .1Λ=-AC C Also find .k A
?????
??--=????
? ??--=201034011)2(,
301121402)1(A A
2.Suppose
????
?
??=?????
?
?=200010000,111
11B y y x x
A
and A ~B.
(1)Find x , y ; (2)Find C such that .1B AC C =-
1.Solution ?
???? ??--????
? ??-????? ??--=Λ=-313
14
131311
001122110101410)1(k
k k C C A ????
? ??---------+--+-=k k k k k k
k
k k k
k k
k
)1(20
)1()1(22)1()1(20)1(31
343
13231
313
132******** (2)A is not similar to a diagonal matrix.
2.Solution 1)x=y=0 ???
?
?
??-=101010101)2(C
Chapter5 Similar Diagonal Matrix(continued)
1. Theorem If n -dimensional vectors )(,,,21n m m ≤ααα are linearly independent, then we can find an orthogonal vector set m βββ,,,21 such that k β can be linearly represented in terms of ).,,2,1(,,,21m k k =ααα The Gram-Schmidt Orthogonalization Process(施密特正交化方法)
111122221111)
,()
,(),(),(),(),(--------
=t t t t t t t t t ββββαββββαββββααβ
That is,
11αβ=
1111222)
,()
,(ββββααβ-
=
222231111333)
,()
,(),(),(ββββαββββααβ--
=
Example1 ???
?
?
??=????? ??=????? ??=001,011,111321αααare linearly independent. Find an
orthonormal set of vectors by the Gram-Schmidt orthogonalization process.
Solution Step1 transform 321,,ααα into an orthogonal vector set by the Gram-Schmidt process
11αβ=
1111222),(),(ββββααβ-=1
232βα-=?
???
? ??-=3231
31 222231111333),(),(),(),(ββββαββββααβ--=2132131ββα--=?
???
? ??-=021
21
Step2 transform 321,,βββ into unit vectors
111ββγ=131
β=?????
?
?
?=3
13131 222ββγ=
263
β=?
????
? ??-=626161
3
33ββγ=
32β=???
?
? ??-=02121
321,,γγγ is an orthonormal set of vectors.
Example2
Solution
2.Definition An n n ? real matrix is said to be an orthogonal matrix if
).,.(,1T T T A A e i I A A AA ===-
3. Properties of an Orthogonal Matrix (1) A is an orthogonal matrix.1±=?A
Proof
(2) A is an orthogonal matrix.1,-?A A T are also orthogonal matrices. Proof
(3)n n n n B A ??, are orthogonal matrices.AB ? is also an orthogonal matrix.
Proof
(4) A is an orthogonal matrix.?The row/column vectors of A are orthonormal set of vectors. Proof
Example is an orthogonal matrix.
4. Properties of the Eigenvalues and Eigenvectors of a Real Symmetric Matrix (1)The eigenvalues of a real symmetric matrix are all real numbers.
(2)The eigenvectors belonging to distinct eigenvalues of a real symmetric matrix are orthogonal. Proof
5.Theorem ?a real symmetric matrix A ,?an orthogonal matrix C , ?
),,,,(211n T diag AC C AC C λλλ ==- where ),,2,1(n i i =λ are the eigenvalues of A .
Example1 Suppose
????
? ??=133313331A
Find an orthogonal matrix C such that AC C AC C T =-1 is diagonal.
Solution A I -λ1
3
3
31
3
3
31
---------=λλλ)7()2(2-+=λλ0= ∴ The eigenvalues of A are .7,2321=-==λλλ
For ,221-==λλthe corresponding homogeneous system is
O X A I =--)2(
that is,
O x x x =???
?
? ??????? ??---------321333333333 A system of fundamental solutions is
????
?
??-=????? ??-=101,01121αα
Next we transform 21,αα into an orthogonal vector set as follows:
11αβ=
1111222),(),(ββββααβ-=?
???
? ??--=11
21
and then we transform 21,ββ into unit vectors as follows:
1
11ββ=
X ????
? ??-=02121
2
22ββ=
X ?????
? ??--=626161 For ,73=λ the corresponding homogeneous system is
O X A I =-)7(
that is,
O x x x =???
?
? ??????? ??------321633363336 ????? ??------→-3363636337A I ????? ??----→990990633???
?? ??---→000990633
????? ??--→000110211?
???? ?
?--→000110101 A system of fundamental solutions is
????
?
??=1113α
Next we transform 3α into a unit vector as follows:
3
33αα=X ?????
? ?
?=3
13131 Let
(),0,,3
16
231612
131612
1321?????
? ?
?-
--==X X X C then C is orthogonal and
).7,2,2(1--==-diag AC C AC C T Example2
Solution
Exercises
1. Suppose .11111????
??
? ??=α Find a set of vectors ),3,2( =k k α such that
),3,2(,1 =k k αα is an orthogonal vector set.
2.Find an orthogonal matrix C and the diagonal matrix Λ
such
that .1Λ==-AC C AC C T
??????
?
?
?=?????
??--=01
001000000100
10)2(,622234243)1(A A 3. Suppose
????
? ??-
-=???
?? ??=3
23
23
1323
132
313232,114P where AP P T Find the eigenvalues and eigenvectors of A .
1.Solution ??????
? ??-
--=??????? ??--=??
??
?
?? ??-=1,01,00113131313212121βββ
2.Solution ????? ??-=Λ?????
? ?
?--=277,0)1(312
343
22312
13
2
2
312
1
C ??????? ??--=Λ??????
?
?
?-
-=1111,0
0000000)2(212
121212
12
12121C 3.Solution );0(,,41313232111≠???
?
? ??-==k k X λ
???
?
? ??+????? ??-===232313231322232,1k k X λλ.)0,(32both not are k k