Algebra Matricial Bequette 1998

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REVIEW OF MATRIX
ALGEBRA
MODULE
2
The purpose of this module is to provide a concise review of matrix operations and linear
algebra.
After studying this module the reader should be able to:
Multiply lnatrices
Find the transpose of a matrix
Find the eigenvalues and eigenvectors of a matrix
Understand concepts of rank and singularity
The major sections of this module arc:
435
In this module we review only the essentials of matrix operation necessary to understand
material presented in the chapters of this textbook. For more detailed explanation:'! and
more advanced concepts, please consult any textbook on matrices or linear algebra.
MATLAB is very useful for performing matrix operations. Module 1 provides a review of
MATLAB.
M2.1
M2.2
M2.3
M2.4
Motivation and Notation
Common Matrix Operations
Square Matrices
Other Matrix Operations
436
M2.1 MOTIVATION AND NOTATION
Review of Matrix Algebra Module 2
In the study of dynamic systems it is comillon to usc l1l<:ltrix BoUllion. The usc of matrix
notation allows the compact representation of a system composed of many variables and
equations.
Matrices are two-dimensional arrays that contain scalar elements that can be either
real or complex. In this module most or our examples involve real matrices. Consider the
following matrix, which consists of n rows and In columns:
r'"
{{II
",.]
A
a2l (/n ([211I
{ll/! {[n2 {fl/Ill
where (lij represents the scalar clement in the ilh row andjth column of matrix A.
It is a good habit to denote the numbers of rows and columns beneath the matrix as
(n,rn) or (n x tn). For example, the following is a 2 x 3 matrix:
A .~ I~ ~ ~I
(2 X 3)
A vec!oris a special case of a two~dimcnsionall1latrix. Normally, the use of the tcnn vec~
tor implies a column vector. The following is an example of a column vector of length :;
(a 3 x I matrix), where Vi is an element of the vector:
A row vector of length three is
W o:c; [-WI 'tV! 'l.03J
The convention of this textbook is to usc lower-case bold leUers to represent vectors and
upper-case bold letters to represent matrices. Unless stated otherwise, it is assumed that a
vector is a column vector.
M2.2 COMMON MATRIX OPERATIONS
M2.2.1 Matrix Addition
Two matrices can be added by simply adding the individual elements of the matrix. The
matrices must have the same number of rows and columns. The operation:
C=A+B
Sec. M2.2 Common Matrix Operations 437
is simply clcmcnt~by-e1cmenladdition to form a new matrix:
(ij :,,"_c aij + b ij
Clearly the order of matrices docs not matter for the addition operation (A + n :::;; B -I- A).
Notice th:'ll the number of columns of B must be equal to the number of rows of A. The cl-
clnenl in the ith row andjth colulllll of Cis:
Consider an 11 x III matrix A and an r x nmatrix n. An ,. x m matrix C is defined as the
product of H times A:
(M2.1)
10
1691 IX12 = 14
II A
(rxn)(nxm)
X
II31 [76 + 10
C :::::
(r x 111)
2
5
Cij = Hi] A Ii + Hi2 A 2j -1- ••• + BinAllj
n
Cij:O:;: 2: BikA kj
k .1
c = I:
For example,
M2.2.2 Matrix Multiplication
Notice that we can view this as an operation on row and column vectors within the matri-
ces. That is, the ith clement in the jth column of C is equal to the scalar value that is ob-
tained from "multiplying" the ith row of B times the jth column of A.
C
(",111)
= H
(r.lI)
A
(1I,1J1 )
Cd
ijth clemenl
Crm
ilh row
i
jlh column
438
[,'or example,
Review of Matrix Algebra Module 2
2
5c ,= [ ~ 1~)j /:11 12
(rows,cols) (3,2) (2,3)
[
7(1) + X(4) 7(2) + X(5)
== 9(1) + 10(4) 9(2) + 10(5)
11(1) + 12(4) 11(2) + 12(5)
(3,l)
7(3) + X(6) j
9(3) + 10(6)
11(3) + 12(6)
[
39
~ 49
59
54
6X
X2
69 jX7
lOS
Clearly the order of the matrices arc important in multiplication. In the example above,
BA is a consistent multiplication, while An is not Even for square matrices (where the
number of rows is equal to the number of columlls), AU -::F- HA in general.
M22.3 Transpose
Let D be 11 transpose of the matrix A, then:
D ~Af (M22)
where (1) represents the transpose operation. The ij clement of Dis theji clement of A
til} = {iji
Columns become rows and rows become columns. For example,
2
5 3..16 A
f ~ [~ ~j
3 6
It caIl he shown that the following property holds:
(ABC)f = CTBTAT
As noted earlier, we normally think of vectors as being column vectors. The transpose of a
column vector is a row vector. Consider an n-dimensional vector x. The transpose of x is xT.
and
Xtl
Diagonal matrices arc commonly used in weighted least squares (regression) problems.
Matrix Q is diagonal if
439
n
2: '/l};Zj
;,,--1
Common Matrix Operations
2 1
r
2 2 Z['lD J Zj'lV2 Z l'l()\
""'·123 ['w, 'W2 IlJtl l Z2~Vl Zz'!D2. 2 2103 Z z1DI/'W3
2,,'101 ZII'WZ Zn?Dj Zll1IJil
Zn
(II.! ) ( 1.11) (11,11)
Notice that multiplication in the opposite order, zw1~ results in ann x n matrix:
The vector transpose is often used to calculate a scalar function of two vectors. [)or CXHrll-
pic, if wand z are column vectors, of length 11, the operation wTz results in a scalar value:
M2.2.4 Diagonal Matrices
Sec. M2.2
For example, Q = diag( I,3, 12) represents the following matrix:
An identity matrix, I, is a diagonal matrix with ones on the diagonal and zeros off'~diagonal.
OJo
12
o
3
o
Qij 'Ii for i ~ j
~ 0 for; i j
IDENTITY MATRIX
[ii ~ I if ; ~ j
oit' ; * j
'fhe 5 x 5 identity matrix is:
[0= diag(l,I,I,J,I)
1 0
o 1
o 0
o 0
o 0
000
000
I 0 0
010
001
440
M2.3 SQUARE MATRICES
Review of Matrix Algebra Module 2
A number of the important matrix operations lIsed in this textbook involve square (num-
ber of rows equal number of columns) matrices.
M2.3.1 Trace
The trace of an n X n (square) matrix is simply the sum of its diagonal clements,
"
tr A :;0: 2: {lii
; 1.
M2.3.2 Determinant
(M13)
One of the most important properties of a square matrix is the determinant. Consider <l
2 X 2 matrix:
A ::::0 la" {/ '.2..1
a 21 an.
The determinant of A is
det A -_0 allan - {f21{f12
Sometimes the det A is written as IAI. An algorthim for finding the determinant of a larger
matrix involves matrix cofactors and is shown below.
M2.3.3 Minors and Cofactors of a Matrix
The matrix formed by deleting the ith row and the jth column of the matrix A is the m;,/or
of the clement aU' c1cnolcdM,j" Consider the 3 x 3 matrix:
o [:::'
au {I'llA an an•.1
(l31 (f-12 aD
The minor of (lj2 of the matrix is obtained by deleting the first row and the second col-
umn. In this case:
ani
o:u __
The detcrminant of the resulting matrix is:
del MI2 0::0 (l21a:n - (1:-Han
The cqj'actor is thc "signcd" valuc of the minor. That is, the cofactor is dcfined as:
c co (- 1)" J del My - q
(M2.4)
(M2.S)
Sec. M2.3 Square Matrices 441
The cofactor e l2 of the 3 X 3 example matrix is:
e\2 cc_-., (~lY+jdctM12 =--= (~1)112(a21a:Bm_{{:l1an) = 112[ll:n + (l:lIa}}
The determinant of a matrix call be found hy expanding ahout any row or column (this ap-
proach is sometimes called the Laplace expansion).
(i) Expanding around any columu j:
"
det A 2: {lijcij
i"\
(ii) Expanding around any row i:
(M2.6)
(M2.7)"det A ::::: 2. {lljCij
j.-.--.- 1
As an example, consider the 3 x 3 matrix A. The expansion around row 1 is
det A = ([nCII + {f12et2 t- ajJen
del A {[" ( .. I) I , I del M" + II 12( -- I)' ' 2 del M 12 + ([ lJ( - I) I ,; del Mil
det A = all (az/l:1,J ~ (f12alT) -- aI2(a21ay~ ~ ({Jl{l2.1) -l-- {{lJ (a21 11:JJ. - ((31022)
M2.3.4 Matrix Inversion
The inverse of a square matrix A,is called A ! and is defined as:
(M2.8)
The matrix inverse is conceptually useful when finding x to solve the following problem:
Ax~y
Multiplying each side (onlheleft) by A', we find:
A-I A x ~
(11 X 11)(11 X 11)(11 X I)
A··' Y
(II X 11)(11 X I)
wbich yields:
Tx~A ' y
or,
where the a(~i()intmatrix of A (adj A) is the transpose of the matrix of cofactofs of A.
ESCOLA DE E,,~CENHARIA
L,,,LI0Tt-:A
(M2.9)
(M2.IO)
I adjAA --,
del A
x ~ A ly
An algorithm for finding the inverse of an 11 x n matrix is known as Cramer's rule:
In practice, this procedure is not used because the computational time is quite large t()r
large matrices. Generally, a method such as Gaussian Elimination or LU decomposition is
used.
442 Review of Matrix Algebra Module 2
EXAMPLE !VI2.1 Inversion of a 2 X 2 Matrix Using Cramer's Rule
A = [all all]
a21 an
A I - adj A
detA
where the cofactofs arc
~ adj A = ,- an
dctA .. ~a21
Cll -- (-l)l+ldetMlI = (1n
e12 = (--I)) t2 dctMj2 = -a:?1
en -~ (-~"1 fl 1 det M21 -- ~ (l12
cn = (-ti l2 det Mn = all
adj A = {c;Jr = [~~: ~~:r' = [~:~
A'
The formula for the inverse of a 2 x 2 matrix should be committed to memory:
A
- [~ b]d
[
d
- b] 1 ~ d...ct A.
a .. detA~E
det A
-b ]det A
....... 0 ...
detA
where de! A 00:; ad - be
That is, ~wap the diagonal terms, take the negative of the off-diagonal terms, and divide
by the determinant.
Sec. M2.3 Square Matrices 443
M2.3.5 Other Issues in Matrix Inversion
A singular matrix cannot be inverted. A matrix A is singular if det A :::: O. Also, a singular
matrix has a rank that is lower than the dimension of the matrix. The rank of a matrix is
the number of independent rows or columns. For example, the matrix:
det A = 4 - 4 = 0
We note that the matrix inverse of a singular matrix does not exist. For this example, note
the division by 0 in the following calculation
has rank = I, because the second row is linearly dependent on the first row. The matrix is
singular, because
-21 1 .. d4 . 0 = undchne-2] I [44. detA = -2AI = I 4
-2
A = I~ ~]
The rank of a square matrix is equal to the number of nonzero eigenvalues. Eigenvalues
of a square matrix arc disclissed next
M2.3.6 Eigenvalues and Eigenvectors
Eigenvalue/eigenvector analysis will bc important when studying dynamic systcms.
Eigenvalucs dctcrminc how "fast" a dynamic system responds, and thc associated eigen-
vector indicates the "direction" of that response. This material is used in Chapter 5.
Eigenvalue/eigenvector analysis can only be perrormed on square matrices. The
eigenvector/eigenvalue problem is:
(M211)
where:
A =
A =
~ =
matrix (square, number of rows::::: number of columns)
cigenvalue (scalar)
eigenvector (vector)
Normally, we will usc the notation ~i to rcpresent the eigenvector that is associated v"ith
eigenvaluc Ai' There are n eigenvalues and n eigenvectors associated with an II X 11 matrix.
The interesting thing about equation (M2.11) is that a matrix times 11 vector is equal
to a scalar times a vector. A scalar multiplying a vcctor docs not change its "direction" in
n-dimensional space, only its magnitude. In this case, a matrix A times the cigenvector ~
yields the eigenvector ~ hack, scaled by A.
Equation (M2.11) can be written as
(AI - A)~ = () (M2.12)
444 Review of Matrix Algebra Module 2
where A must be a scalar value for which Al --- A IS singular, otherwise ~ :;;;; 0 (the trivial
vector). A requirement for AJ - A to be singular is dctCAJ ~ A) :;;;; 0, where the notation
det(AI - A) is used to represent the dctcnninant of l\!- A.
EIGENVALUES
The eigenvalues of an n X n matrix A arc the n scalars that solve:
det(l\! - A) •.. 0 (M2.13)
Equation (M2.13) is also known as the characteristic polynomial for the matrix A~ the
eigenvalues arc the 1'001,0;; of the c1Ulractcristic polynomial. For an 11 X 11 matrix A, the char-
acteristic polynomial will he nth order, so there will be II roots (n eigenvalues).
EIGENVECTORS
A
The corresponding 11- eigenvectors (~) can be found from
A ~i l\j ~i
~i is the eigenvector associated with the eigenvalue Ai-
For example, consider the general 2 x 2 matrix:
1
1I'.l (Jizi
(In (/22_
(M2.14)
l\I-A
1
l\-II 11
.. ~(/Zl
det(l\I - A) = (l\ - a,,)(l\ - II,,) - all "21 =. 0
det(l\I- A) = l\' - [1111 + lin] l\ + {alia" - a2l1l,,{ ~ 0
Equation CM2J5) can also be written as:
det(l\! - A) ~ l\' - triA) l\ + det(A) ~ 0
where:
triA) = all + a22
dct(A) = allan - anlln
Equation (M2.16) is the characteristic polynomial for a 2 x 2 matrix.
(M2.15)
(M2.16)
It can easily he shown that, if tr(A) < 0 and det(A) > 0, the roots (eigenvalues) of (M2.16)
will be negative. Also, if det(A) '= 0, then one root (eigenvalue) will be zero.
The reader should derive the following characteristic polynomial for a 3 x 3 matrix.
(see student exercise 7).
Sec. M2.3 Square Matrices 445
The characteristic equation for a J x 3 matrix is:
det(Al- A) = A' - tr(A) A' + M A- dCl(A) = ()
where:
tr(A) = (/Il + (/" + {/11
M c~ det Mil + del M" + det M11
:0::: "ntr:,>:,> -" (lJ211n + {lllaJ3 ~ {fJl((jJ + a lJ a 22 - {{12(f21
det A =0 (l11(a22{f:n - anil lz ) ~ {[12(an llJ:. - ((?,j(ln) + {/u(a2111:12 ~ (l:\1(fn)
(M2.17)
A
The ROllth stability criterion (Chapters 5 and 9) can be Llsed to find the conditions on the
polynomial coefficients that will yield negative roots (eigenvalues). The necessary condi-
tion is that all of the polynomial coefficcnts arc positive. The necessary condition is then
tr(A) < 0, M > 0, and dct(A) < O. The reader can usc the Routh stahility criterion to deter-
mine the sufficient conditions for negative roots.
EXAMPI..EM2.2 Eigenvalue/Eigellvector Calculation
Consider the following matrix:
~ I~ -\ I
dCl(,.I~A) ~ A' -[2~ 1[,. 1-[2(-1)-2(1)[ ~ IJ
A! ,.~4cO
F'rolll the quadratic equation:
I I - 4( 4) I VI7
A +2 2 2
so,
and
A, ~ 2.5616
The first eigenvector is found from (M2.14):
where we have lIsed the notation:
(M2.1X)
446
Applying equation (M2.1S),
I~
Review of Matrix Algebra
Iv I1.5616 11'V21
Module 2
So the following fWD equations must be satisfied:
2V11 +V21 1.56J6v11
2V11 -- 021 = - 1.5616 'On
which yield:
1J21 3.561 5 ~/)]l
If we wish, we call arbitrarily select VII 1.0 and '/)21 = 3.56J5
1
II" I [ 1.0 I~, cc c
}}n--- 3..l6!5 1M2. J9)
Most complltcl:pal:kag~:"\~il] report cigyllycctors that have a unit norm (length:o:: J). If we divide
1M2. J 9) by V(1)" V+ (1',,)2 VI-:, 12.6841 3.6992, we rind that:
o.W)})
0.9628
An (~qllivalent solution is:
1
- O.2703j.
0.9628
The reader should show that:
leads to:
~2 - I. '~I !]tv 10.8719]0.4896
Eigenvector problems arc easily solved used the eig [unction in MA'T'LAB (sec Mod-
uleJ), as shO\vn below.
a =
2 1
2 -1
v =
d
0.8719
0.4896
2.:')616
o
-0.2703
0.9628
o
-1. 5616
Sec. M2.3 Square Matrices 447
Column i in the v matrix is the eigenvector associated with the ilh eigenvalue (which is
the ith diagonal clement ill the d matrix). The trace and determinant can also be f(lUnd
using MATLAB commands:
» trace(a)
an~; -
1
» det (a)
ans
··4
M2.3.7 The Similarity Transform
The similarity tran.~I()nn is uscful for understanding the dynamic behavior or linear sys-
[ellls (sec Chapter 5). Recall the eigenvector/eigenvalue problem for a 2 x 2 matrix:
(M2.20)
(1112.21 )
Notice that equations (M2.20) and (M2.2l) can be written in the following form:
AV VA
where V is the matrix of eigenvectors:
(1112.22)
(M2.23)
where:
A is the matrix of eigenvalues:
1/)".1 1/)"11)21. and ~2 = v)~
;\
°1A} .. (1112.24)
Multiplying (M2.22) on the right side by VI we find
A' V A V I (111225)
r~quation (M2.25) will he useful when developing analytical solutions for sets of Iincar
differential equations, and can be used for any n x II (square) matrix.
448 Review of Matrix Algebra Module 2
M2.4 OTHER MATRIX OPERATIONS
M2.4.1Vector Norms
It is quite natural to think of vectors as having a "length" property. This property is called
a vector norm. The most common norl11 is the Euclidean norm. In two-dimensional space,
the Euclidean norm is:
II xII = V~; +~l
Sometimes the Euclidian norm is called the 2-nonn, JJ x lb.
In n-dimensional space,
II X II, =Vx; + xi + ....•~.\;' = ~± xJ
,coj
notice that the sum of the squares can be written as xTx , lhalis:
and the 2-norm is then:
IlxI12=~
M2.4.2 Orthogonality
A column vector w is orthogonal to a column vector z if wTz = 0.0.
For example, consider the 2-D vector l 1 "Iw= -O.S
and the 2-D vector
z (0.5,1)
w (1,-0.5)
xTz = [l ~O.Sl [O;S] = I(O.S) + (-O.S)1 .~ OJ)
Notice that orthogonal vectors arc "perpendicular" to each other, as depicted below.
Then
M2.4.3 Normalized or Sl'aled Vectors
Many times, we will scale vectors, so that they are !1onna/ized. A vector is normalized if
it has a norm (length) of 1.0. Normalization is pcrformed by multiplying each c1ement_ or
thc vector by I/vector norm. For a givcn vector, x, the scaled vector, U is u:::: X l/~
EXAIVIPLE M2.3 Vector Normalization
Consider the vector x T = [l 21:
Then,
And,
XIX [I 2JI~I~1+4c5
The normalized vector is
F I~I ~ ~ [£;]
A vector norm 011 RII is a function I1II :RII 4 R satisfying:
1. x:;t: 0 4
2. II (XX II
3. II X j YiI .,;
II x II > 0
I (X III xii
II x II + II y II
Where CI.. is a scalar
'rriallglc inequality
Common Vector Norms
"
I. II X Iii ~ 2: IXi I
j······l
2. II X II, ~ (~ IXi If'
3. II x II,. (~ Ix, I") 1/,.
4. II x"" ~ max Ix, I
l-norm (sum of absolute values)
2-norm, Euclidean Norm
p-nonn
oo-norm
Notke that the p-norm includes all other vector norms
EXAMPLE M2.4 Comparison of Norms
Xl ~ [I -2 31
II x Ii, III I 121 + 131 ~ 6
!I X II, ~ (iJI' I 12/' + Im'/2 ~ 3742
Ilxt ~ max 1111.121, 1311 ~ 3
Ilxll,,, ~ (IJI'" + 1_21 '0 + 131 10)1/10 3.005
Notice that. as p .., =.11 X II" --> II x IL.
450 Review of Matrix Algebra
SUMMARY
Module 2
In this module we have provided a concise review of matrix operations. This material
should be sufficient for most of the matrix operations used in the textbook. It is partiCLI-
larly important that the reader understand:
Matrix notation
How to multiply matrices
That an n X 11 matrix has n eigenvalues, which arc the 11 roots to the characteristic
equation: det(~[ - A) ~ 0.
That a matrix A is singular if the dct(A):::: O. A singular matrix docs not have an in-
verse. A singular matrix is not full mnk.
That the rank of a square matrix is equal to the number of nonzero eigenvalues
STUDENT EXERCISES
For problems 1-3, usc x" [~]
I. Find x'.
2. l<ind xTx.
3. Find II x Ib
4. Perform the indicated matrix operations for the following matrices:
f
3
A ~ 1
a. FindC ~ All
b. Find [) ~ IliA
c. Find E A 'll
B~ [7
9
6
5
5. Solve for x as a fUllction of.y where:
6. Find the determinant of:
v ~ [I
. 2 1] x
-1
7. Derive the characteristic polynomial (del (1\.1- A)) for a general 3 x 3 matrix:
[""
1112
11
11
]A ... {/21 an ((21
([-\I {lJ2 {lJl