2026 Prof.Jiang@ECE NYU 214 Lecture V • The higher dimensional case of general
real symmetric matrices • Extension and Applications 2026 Prof.Jiang@ECE NYU 215 The General Case We already proved the result with 2.
,
let us assume that for each integer 1 ,
we can find an
matrix
which reduces real symmetria
matrix
to the diagonal fo ortho c g rm:
na
o l k k ij k k N By induction k N O A a 1
0 0 T k k k k O A O 2026 Prof.Jiang@ECE NYU 216 The General Case: Goal 1 1 ( 1) ( 1) 1 1 1 1 1 We want to find an orthogonal matrix
of order 1, which reduces a real symmetric matrix
to the diagonal form:
0 0 N N ij N N T N N N N O N A a O A O 2026 Prof.Jiang@ECE NYU 217 Systematic Procedure 1 ( 1) ( 1) 1 2 1 1 1 1 Let us name the rows of
as:
take an eigenvalue
and its associated eigenvectornormalized
.
N ij N N N N A a a a A a and x 2026 Prof.Jiang@ECE NYU 218 Systematic Procedure (cont’d) 1 1 1 1 2 1 1 By means of the Gram-Schmidt orthogonalization process, we can form an orthogonal matrix
whose first column is the
:
,
,
,
Then, as shown in C give ase 2, it holds: n N O x y O y y y N 1, 11 12 1 1 1
0
,
0 N T N N N N N b O A O A A b 2026 Prof.Jiang@ECE NYU 219 Exercise Can you prove the above identity? 2026 Prof.Jiang@ECE NYU 220 Answer 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 11 1 2 1 1 1 1, 1 Carrying out the multiplication, we have , ,
, ,
, ,
, , N N N N N N N N N N a x a x A O a x a x x a x a x x a x a x 2026 Prof.Jiang@ECE NYU 221 Answer (cont’d) 1, 1 1 1 12 1 1 1 Since
is an orthogonal matrix, it follows that
0
,
0 N T N N N N N O b O A O A A b 2026 Prof.Jiang@ECE NYU 222 Comment 1 1 1 1 1 1 1 1 1 Furthermore, since
must be symmetric, we have 0, 2,..., 1,
and
is symmetric. Thus, 0
0 0
,
. 0 T N j N T T N N N N O A O b j N A O A O A A A 2026 Prof.Jiang@ECE NYU 223 Comment 2 1 1 1 1 2 3 1 1 Given the identity 0
0 0
,
0 we conclude that the eigenvalues of
be ,
,
,
,
the remaining eigenvalues of . T T N N N N N N N O A O A A A A must A 2026 Prof.Jiang@ECE NYU 224 Systematic Procedure (cont’d) 1 1 1
, there is an orthogonal matrix
which reduces
to diagonal form. Form the ( 1)-dimensional matrix 1 0
0 0
0 Clearly,
is also orthogonal, i.e.,
N N N N T N N By induction O A N S O S S 1 . NS I 2026 Prof.Jiang@ECE NYU 225 Systematic Procedure (cont’d) 11 1 1 1 1 1 1 1 1 1 1 1 1 1 It can be directly checked that
or, equivalently,
with . 0 0 0 0 T T N N N N T N N N N N N S O A O S O A O O O S 2026 Prof.Jiang@ECE NYU 226 Formal Statement of the Main
Result 1 1 Let
be a real symmetric matrix. Then, it may be transformed into a diagonal form by using an orthogonal matrix
so that
where
are the eigenvalues of . 0 0 n n T n n i i A O O AO A 2026 Prof.Jiang@ECE NYU 227 Test for Positive Definiteness A necessary and sufficient condition for a real symmetric matrix
to be
is that all eigenvalues of
are po
sitive. positive de A finiteA 2026 Prof.Jiang@ECE NYU 228 Indeed, 1 2 1 Recall that a real matrix
is
if 0,
, 0. , ,
where
,
where
So, the equivalence property follows readi posit ly ive definite T n T T T i T T i n n i i i A x Ax x x Then x Ax x O O x diag y y y O x y y . 2026 Prof.Jiang@ECE NYU 229 Repeated Eigenvalues 1 As shown previously, if a matrix
has
eigenvalues, then its associated eigenvectors
are linearly independent.
What if
has a repeated eigenvalue
of (algebraic) multiplicit ? : y
Ques A t distinc A i t ons k Are there always
linearly independent eigenvectors? k 2026 Prof.Jiang@ECE NYU 230 Comment As shown in , the answer is generally negative for a real matrix which is symmetric. However, for a real symmetric matrix, we can always find
linearly independent eigenvectors for
repe Lecture
IV any not k ated eigenvalue of multiplicity .k 2026 Prof.Jiang@ECE NYU 231 Indeed, 1 1 1 1
an orthogonal matrix
such that
where ,
, 1,..., . ,
the first
columns of
are of course linearly independent, and are eigenvectors associated with
0 0 n k i O AO O i k n Now k O . 2026 Prof.Jiang@ECE NYU 232 In addition, 1 1 Any other eigenvector
associated with
of these
vectors. In fact, ,
with
the th column of . 0, 1,
,
because these eigenvectors
are linear combina
orthogonal w o ith
ti n n i i i i i i y is k y c x x i O c i k n x ,
,
1 . (
Lecture IV) jx y j k see 2026 Prof.Jiang@ECE NYU 233 Special Case of Cayley-Hamilton Theorem 1 1 As a direct application of the diagonal canonical
form, we have
:
0, where det ,
: A n n n i A i i n n n i A i i Any real symmetric matrix satisfies its own characteristic equation A I A A A A 0,
with .A I 2026 Prof.Jiang@ECE NYU 234 Application: Solving Differential Equations 1 1 1 Solving for the solutions of
an
problem for
where
is a canonical form of
under a nonsingular transformation e
:
s asier o that
a c c c x Ax boils down to y A y A A P x y P x A P AP nd .x Py 2026 Prof.Jiang@ECE NYU 235 Exercise 1 Solve the following initial-value problem: 1 3 1 ,
0 . 3 1 0 . ,
?,
0. x t x t x i e x t t 2026 Prof.Jiang@ECE NYU 236 Exercise 2: Extension to Difference
Equations 0 1 1 2 Find an explicit expression for ,
0,1,..., given that
1,
2 and
,
2,3,... n n n n x n x x x ax x n 2026 Prof.Jiang@ECE NYU 237 Hint 1 2 1 1 1 Rewrite the second-order difference equation as: 0 1
,
2,3,... 1 Equivalently,
,
1, 2,... with
: . n n n n n n n n n x x n x a x A n x x 2026 Prof.Jiang@ECE NYU 238 Another Extension When does there exist an orthogonal matrix
which simultaneously reduces two real symmetric matrices ,
to diagonal f
o : rm? Questi O B n A o 2026 Prof.Jiang@ECE NYU 239 Motivational Problem Solve the 2nd-order differential equation:
( ) ( ) 0,
where ,
are
matrices. Note that such equations often occur in
mass-spring probl symmetric ems. n n n Ax t Bx t x A B 2026 Prof.Jiang@ECE NYU 240 Basic Result A necessary and sufficient condition for the existence of an orthogonal matrix
such that
is that
and
commute, i.e. . T i T i O O AO diag O BO diag A B AB BA Note: See Section 1.3 of the 2013 textbook of Horn and Johnson for extensions
to more than 2 matrices. 2026 Prof.Jiang@ECE NYU 241 Proof of the Necessity Clearly,
and
commute, because
is orthogonal. T T i iA O diag O B O diag O O 2026 Prof.Jiang@ECE NYU 242 Sufficiency: Sketch of Proof
Either
or
has distinct eigenvalues. Assume that
has distinct eigenvalues. Then, . So, ,
if nonzero,
is an eigenvector too,
for the same
eigenvalue . In other words,
1: A B A Ax x A Bx B Ax Bx Bx Bx Case is a multiple of . As a result, ,
for each pair ,
.i i ii i x Bx x x This equality of course also holds if Bx=0. 2026 Prof.Jiang@ECE NYU 243 Sufficiency: Sketch of Proof 1 2 ,
we observe that ,
have the same
eigenvectors ,
1 . Thus, we can define the orthogonal transformation matrix
as f
1 (con ollows:
, , ) , t'd i n Now A B x i n O O x se x x Ca 2026 Prof.Jiang@ECE NYU 244 Sufficiency: Sketch of Proof 1 1 1
repeats
times associated with (linearly
independent/orthonormal) eigenvectors , , . ,
using previous computation, we have
,
1, 2, , . In addition, , ,
2 :
i k k i j ij j j j ij k x x Th C en Bx c x i k x Bx ase c Bx x 1 . ,
consider the linear combination . i ji k i i i c Now a x 2026 Prof.Jiang@ECE NYU 245 Sufficiency: Sketch of Proof 1 1 1 1 1 1 1 1 1 1 1 We have ( ) . Thus, if we choose
so that
,
1, 2, ,
2 (cont'd) :
, 0 then we have
k k k k k i j j i i ij ij i i i j j i i k ij i j i k k i i i i i i B a x a c x c a x a c a ra j k r I C a B a x r a Case x 2026 Prof.Jiang@ECE NYU 246 Sufficiency: Sketch of Proof 1 1 1 1 1 1
implies
is an eigenvalue of , associated with eigenvector . On the other hand,
is an eigenvalue of
associated with eigenvector
2 (cont'd) : k k i i i i i i k i i i ij Ca B a x r a x r se B a x r C c a c .iol a 2026 Prof.Jiang@ECE NYU 247 Sufficiency: Sketch of Proof 1 1
If
is a -dim. orthogonal transformation reducing
into a diagonal form, then
is an orthonormal set with each
being
eigenvector for both
2 (end) : comm n and .
o k k k k i T k C z z T x x z A B lef Case t
as an exercise Exercise 3 2026 Prof.Jiang@ECE NYU 248 Show how to transform the following matrix into a canonical diagonal form, by means of an
orthogonal matrix: 1 2 0
2 1 0 0 0 1 M Normal Matrices: A generalization of real symmetric matrices 2026 Prof.Jiang@ECE NYU 249 * * Definition: A square matrix
is norma said to be ,l if . A AA A A
If
is
and
is a scalar, then
also is normal.
If
is
and ,
then
also is normal.
Every unitary matrix is normal.
Every real symmetric or skew-symme normal tric m normal atrix is normal.
A A A B A B Every Hermitian or skew-Hermitian matrix is normal. Exercise 4 2026 Prof.Jiang@ECE NYU 250 Let ,
be constants. Show that
is normal
and has eigenvalues . a b a b b a a ib Exercise 5 2026 Prof.Jiang@ECE NYU 251 11 12 22 11 22 A matrix
is
if = . Show that a
matrix
is conjugate normal 0 if and on block-upper-tr ly if its diagonal blocks ,
are conjugate
norma conjug iangular l, ate normaln nA AA A A A A A A A A 12 and 0. In particular, an upper triangular
matrix is conjugate normal iff it is diagonal. A See the text (2nd ed.) by Horn-Johnson, p. 268. 2026 Prof.Jiang@ECE NYU 252 Homework #5 1 2 3 1 2 3 1. Using the Gram-Schmidt process find a set of mutually orthonormal vectors , , , based on: 1 0 0 0 1 0
,
,
. 0 0 1 1 1 1 u u u x x x 51作业君版权所有
