2026 Prof.Jiang@ECE NYU 117 Lecture III Key issues: • Eigenvalues, eigenvectors and the
characteristic polynomial of a square matrix • Similarity 2026 Prof.Jiang@ECE NYU 118 Eigenvalue and Eigenvector t Given a square
matrix ,
the set of s
are such that the
, or equivalently 0 characteri
stic equati
r on ei
genv l eig t enva o ec has a nonze o so u i n 0. Such a vector
is called an lue
n n n A Ac c I A c c c associated with eigenvalue . or Definition: As a result, for any eigenvalue,
det 0,
.I A A 2026 Prof.Jiang@ECE NYU 119 How to compute an eigenvalue? 1 1 1 characterist The eigenvalues ic polynomia
of a matrix
are the roots of its : det
l
. n n n n n n A I A So, in total there are n eigenvalues (possibly repeated and/or complex-valued). They are the roots of the characteristic polynomial of A: 2026 Prof.Jiang@ECE NYU 120 Comment on Characteristic Polynomial 11 1 , det 0 implies, by means of permutation: det 0. Indeed n n n n I A I A 2026 Prof.Jiang@ECE NYU 121 Comment 1 For any given eigenvalue λ, the eigenvectors
are
1) nonzero solutions to the homogeneous
equation: (λI - A)c=0. 2) non-unique. Clearly, any vector of the form µ x c for a
nonzero scalar µ
is still an eigenvector. 2026 Prof.Jiang@ECE NYU 122 Comment 2 For any given eigenvalue λ, the associated
eigenvectors c are nonzero elements of the
null-space of (λI - A), i.e.:
So, the maximum number of linearly
independent eigenvectors is equal to: n – rank(λI - A). ,
0.c Null I A c 2026 Prof.Jiang@ECE NYU 123 An Example 1 2 1 2 1 0 For the identity matrix ,
the eigenvalues 0 1 are 1,
for which two
eigenvectors are: 1 0
,
. 0 1 Othe linearly indep r choices of i end ndependent eigenvectors are:
e
t
n A c c 1 21 3,
. 2 1 c c 2026 Prof.Jiang@ECE NYU 124 A Useful Observation A square matrix
of dimension
is singular
it has a zif and on ero eigenly i vaf lue. A n 2026 Prof.Jiang@ECE NYU 125 Proof
Assume that
is singular. Then, there is a nonzero solution 0 to the equation (*)
0,
or equivalently, 0 . So, 0 is an eigenvalue. If 0 is an eigenvalue,
then t : he : A x Ax Ax x Necessity Sufficiency re is an eigenvector 0,
which is a solution to (*).
Thus,
must be a singular matrix. x A 2026 Prof.Jiang@ECE NYU 126 Exercise Find the eigenvalues and the associated eigenvectors of the following matrix 1 1
. 1 1 M 2026 Prof.Jiang@ECE NYU 127 More about Eigenvalues
A real square matrix can complex have eigenvalues. 1 2 For example, the eigenvalues of 0 1
1 0
are ,
.j j 2026 Prof.Jiang@ECE NYU 128 More about Eigenvalues
A real square matrix can multiplehave eigenva . lues 1 2 1 2 For example, the eigenvalues of 1
0 1
are 1 for any . We say that 1 is an eigenvalue of multiplicity 2. In this case, the characteristic polynomial is det 1 . a A a I A 2026 Prof.Jiang@ECE NYU 129 More about Eigenvalues 1 21 2 1
More generally, the characteristic polynomial of
a matrix
takes the form
det
, ,
are
an eigenvalue of (algebr different, wi aic) m th , ultiplicit
r n n m m m r r i i i A I A where m n 1 1 . When
and 1,
the matrix
is y
dis said to
have eiget nvalinc ue
.t s n n i i i r A m n m m 2026 Prof.Jiang@ECE NYU 130 More about Eigenvectors 1 1 2 1
Back to the example of . It has an eigenvalue 0 1
1 of (algebraic!) multiplicity 2, for any .
For 0,
the associated eigenvectors ar distine : 1 0
Cas
e 1: ,
0
ct
a A a a c c . 1
2026 Prof.Jiang@ECE NYU 131 More about Eigenvectors 1 1 e Ca e 1
Back to . It has an eigenvalue 1
0 1
of (algebraic) multiplicity 2, for any .
For 0,
there is
distinct
1
eigenvector:
0 . .,
only one all other eigenv s
2: a A a a c i e 1ctors take the form ,
for some
scalar 0. r c r Its “geometric” multiplicity is 1 2026 Prof.Jiang@ECE NYU 132 Exercise Compute the eigenvalues of the following matrix
1 1 1
1 1 1 . 1 1 1 Can you find the eigenvectors associated with each eigenvalue? M 2026 Prof.Jiang@ECE NYU 133 Exercise For any matrix ,
1) can any eigenvalue of
be complex? 2) can the largest eigenvalue of
be negative? n n T T A A A A A 2026 Prof.Jiang@ECE NYU 134 A General Result 1 1 Assume
has
eigenvalues ,
...,
. ,
(1)
must have
eigenvectors
,
,
. (2) In addition, each eigenv li ec nearly independen d tor
associated with
is
istin apart f ct t unique n n n n j j A n Then A n c c c rom a nonzero scalar multiplier. 2026 Prof.Jiang@ECE NYU 135 Proof of Statement 1 1 1 Let 0,
,
0 be eigenvectors satisfying
,
for 1, 2, , . We prove the statement by contradiction. Assume that
are linearly dependent. Let
be the le
positive integer suas cht n i i i ni i c c Ac c i n c k n 1 1 2 1 2
that
of the 's are dependent. Without loss of generality, assume that
are dependent, th not aat ll is,
zer
,
o
0. ki ii k k k c c c c c 2026 Prof.Jiang@ECE NYU 136 Proof of Statement 1 (cont’d) 1 2 1 2 1 1 1 1 1 1
,
0. Thus, 2 and
0 (otherwise, contradiction with
being the least). ,
multiply the eq. b not al y
leads to:
0 which,
l zero all
in turn, i k i k i k k k k k k c c c k k Now A I c c 1 1 mplies that
are dependent. A contradiction. i i k c 2026 Prof.Jiang@ECE NYU 137 Proof of Statement 2 1 1 1 We must show the " " of :
,
0
,
with 0. As it was proved in statement (1),
are linearly independent and thus form a basis. So uniqu ,
. Multiplyi eness ng t i i i ni i n n i i c Ac c c c c c c c cc 11 1 he above eq. by
gives:
0 Therefore:
0,
. In other words, as wished 0 ,
. i n i n i k i n i A I c c k c i c 2026 Prof.Jiang@ECE NYU 138 Corollary 1 2 1 Under the above conditions, define matrix
which is nonsingular & implies
. In this case, we say t similar
similar hat
is
, while
is it y maa o t t
n i i P c c c P AP diag A diag P .
Denote .
is called diagonali rix zable" "iA diag A 2026 Prof.Jiang@ECE NYU 139 Remark 1 As we will see in Lecture IV,
is
a canonical form for
1) the class of matrices having distinct eigenvalues; and
2) the class of real symmetric matrices,
which may have repeated eigenvalues. idiag 2026 Prof.Jiang@ECE NYU 140 Remark 2 1 1 1 1 1 Indeed, ,
such that . It then follows that
de Any two similar matrices
and
must have the
t det det det det det det ,
because det det d same eigenvalues. A B A B P B P AP B I P AP I P A I P P A I P A I P P 1et 1.P P 2026 Prof.Jiang@ECE NYU 141 Questions Are you ready for some tricky questions? 2026 Prof.Jiang@ECE NYU 142 Question 1 3 3 3 If the set of all eigenvalues of , or the
1, 2,spectrum 3 ,
what is ? A A A A 2026 Prof.Jiang@ECE NYU 143 A General Result 0 0 0 For any polynomial , and any
matrix ,
denote ( ) ,
with . If ,
is a pair of eigenvalue and eigenvector of , then ( ),
is a pair of eigenvalue
and eigenvector of ( ) k i i i k i i i p t a t n n A p A a A A I A p x x p A . (Its proof is left as an exercise.) “Matrix Polynomial” 2026 Prof.Jiang@ECE NYU 144 Question 2 2For any
matrix , that is, , what are the possi idempo ble ei ten gen t values? A A A Idempotent Matrix: 2x2 case 2026 Prof.Jiang@ECE NYU 145 2 2 If
is idempotent, then , ,
implying 0,
or 1 ,
implying 0,
or 1 . a c A b d a a bc b ab bd b d a c ca cd c d a d bc d Examples: Idempotent Matrix 2026 Prof.Jiang@ECE NYU 146 0 (1)
is idempotent, if , 0,1. 0 1 cos sin1 (2) . sin 1 cos2 a A a d d A 2026 Prof.Jiang@ECE NYU 147 Answer An idempotent matrix can only have 0 or 1
as its eigenvalues. 2026 Prof.Jiang@ECE NYU 148 Question 3 A
matrix
is such that 0 for a positive integer . Such a smallest
is called the
.
What are the eig nilpotent ind envalues of
ex of nilp a nilpo o t t ent matrix ? ency qA A q q A 2026 Prof.Jiang@ECE NYU 149 Answer All eigenvalues of a nilpotent matrix are 0. Indeed, if ,
0,
then, using 0,
we obtain 0 which, in turn, implies 0. q q Ax x x A x Examples: Nilpotent Matrix 2026 Prof.Jiang@ECE NYU 150 0
* *
(1)
* 0 * 0
0 0 5 3 2 (2) 15 9 6 10 6 4 M M Equivalence Relation: Nilpotent Matrix 2026 Prof.Jiang@ECE NYU 151 The following statements are equivalent: (1)
is nilpotent. (2) The minimal polynomial of
is ,
for some . (3) The characteristic polynomial of
is . (4) The only eigenvalue of
is 0. n n q n M M s q n M s M 2026 Prof.Jiang@ECE NYU 152 Exercise Compute the algebraic and geometric multiplicities of the eigenvalue 2 for the matrix 2 1 0
0 0 2 0
0
0 0 2
1 0
0
0
2 A 2026 Prof.Jiang@ECE NYU 153 Useful Identities about Matrix Eigenvalues 1 1 For any
matrix ,
we have
( ) ,
det . n i i n i i n n A trace A A See p.42 of (Horn-Johnson, 1st ed., 1985), or p. 50 of (Horn-Johnson, 2nd ed., 2013). Similarity 2026 Prof.Jiang@ECE NYU 154 Two matrices A and B are said to be similar, if
1 ,
for some invertible matrix
Notatio . n:
B P AP P A B 1 The set of all similar matrices to a given square matrix :
:
is invertible Similar matrices are just different basis representations of
a single li No near map te ng. :
pi A S P AP P Similarity: Physical meaning 2026 Prof.Jiang@ECE NYU 155 1 1 1 2 1 1 1 1 1 1 2 Let :
be a linear transformation, and
, , ,
, ,
be two bases for . Denote , ..., ,
with ... . ,
by linearity,
... For any basis
of , n n n n nB n n j T V V B v v B w w V x col x v v Then Tx Tv Tv B V Tv 2 2 2 1 1 1 , ..., , , ..., . j njB n j j ij nB B nxn j col t t Tx Tv t col Similarity: Physical meaning 2026 Prof.Jiang@ECE NYU 156 2 2 1 1 1 2 1 2 , ..., . It is important to note that the matrix depends on
and the choice of the bases
and ,
but
. Define the -
as:
not n j j ij nB B nxn j ij nxn Tx Tv t col t T B B x B B basis representation of T 2 1 2 2 22 1 1 1 1 1 1 2
1
, ..., ,
= , . For the special case when ,
is called the
representatio .n of
B ij nB B Bnxn BB B B B B T t Tv Tv B T So Tx T x x V B B T Similarity: Identities 2026 Prof.Jiang@ECE NYU 157 2 11 1 2 2 2 1 2 2 1 12 1 1 2 1 1 22 1 2 1 For the identity linear transformation , , it can be shown that ,
,
=
. In other words, ,
where ,
,
. B n B nB B B B B B B B B BB B B B B B BB B B Ix x x V I I I I I I and T I T I B P AP P I A T B T For a proof, see (Horn & Johnson, 2nd ed, 2013, page 40). 2 1-
change of basis matrixB B Exercise 2026 Prof.Jiang@ECE NYU 158 1
an invertible matrix
such that 1 1 1
0 2 2 0 0 3 is diagonal. Find P P P Exercise 2026 Prof.Jiang@ECE NYU 159 1 Let ,
be similar:
,
for some . For any eigenvector
of ,
show that
is an eigenvector of . n n n n n n A B A P BP P x A Px B 2026 Prof.Jiang@ECE NYU 160 Homework #3 1. Give all the solutions of the system 1 2 3 10 13
4 5 6 .11 14 7 8 9 12
15 2. Prove that the following eq. has no solution: 1 3 1
. 2 6 3 x x 2026 Prof.Jiang@ECE NYU 161 Homework #3 1 2 0 1 2 1 2 3. Find a least-squares fit
for the data: 1 1 0 0 2 0
,
,
. 0 3 0 0 4 1 b x x a x a b a a 2026 Prof.Jiang@ECE NYU 162 Homework #3 4. Find independent eigenvectors for 1 2
. 3 1 1
Can you express
as a linear combination 2
of these eigenvectors of ? A x A 51作业君版权所有
