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Unformatted text preview: Econ 5000 II. Linear Algebra Part 11: Linear Algebra Fundamentals 1. Elements 9 Scalar: a simple dimensionless element, (i.e. 4 or 7 etc.). 9 Vector: a one dimensional array of numbers usually denoted as
follows: {1,2,3}. There are two types of vectors, row vectors and column vectors. The vector x = hg,xw...,xﬁ is a (lxn) row vector.
x
1
x
X = :2
x
m is an (mxl) column vector. 9 matrix: a two dimensional array of numbers, the matrix A can
be written as: an an " am
A— 3.21 a22 a2n
aml a“12 aInn
or
A: {aﬁ} i = 1,...,m
j = l,...,n where i is the row index and j the column index. Econ 5000 II. Linear Algebra 6 A vector is a matrix with only one row or column.
0 The matrix A is said to be square if m = n. 0 Two matrices, A and B are equal (A = B) if and only if aﬁ
for all i,j. u
{55' 2. Matrix Operations Addition and Subtraction: If A and B are two matrices of the same
order (i.e. they have the same number of rows and columns) they can be added or subtracted. Addition and subtraction take place
element by element. Ex. C = A + B implies, {cij = aij + bﬁ} i = l,2,...,m;
j=l,2,...n.
A=[an an] B=[b” bu] 321 322 b2! bzz C=A+B= a11+b11 a12+b12
a2144321 azz+b22 D=A—B=[a” "bu 312—1312] a2: _b21 a‘22 'bzz Scalar Multiplication: If k is a scalar and A is an an matrix
then each element of A is multiplied by the scalar k. kA = {kaij} i = l,...,n; j = l,...,m Inner Product: Consider two vectors, Aniand er (A is a row vector ands B is a column vector) The inner product, AB is a
scalar: A«B =a,bI +a2b2 +...+anbrl
also known as the dot or scalar product. Extremely useful for accumulating sums. For example, XY=zxy Econ 5000 II. Linear Algebra
Cross Product: The cross product BA is a an matrix, C, such that
c“ =b a. 13 :L 3.
matrix multiplication: Consider two matrices A an and B qu. O For AB or BA to exist, the matrices must be conformable. 0 The product AB of these two matrices exists if and only if n =
p. O The product BA of these matrices exists if and only if m = q In general ABiBA For two conformable matrices A(mxn) and B(nXp). The matrix C = AB
is found by succesiVely taking the dot product of the rows in A by
the columns in B. Cij is the dot product of the ith row of A and the
jth column of B. ex.
3 a
A = ' 11 12 B :[bu blZ b13
321 a22 b2. b2: b23
C=AB= all 12 2 11 12 11 12 a21b11 +322b21 a21b12+a22b22 a21b13"""122b23 The product of a HMn and a nXm matrices is a HMm matrix. The (cross) product of a mxl matrix (ie., a column vector) and a le
matrix (ie., row vector) is a me matrix. The (inner) product of
an matrix (ie., a row vector) and a nxl matrix (ie., a column vector) is a 1x1 matrix (ie., a scalar). Econ 5000 II. Linear Algebra 3. Special Matrices O A matrix is diagonal if it is square and a”. = 0 Viij O A matrix is triangular if an = 0 V j>i (lower triangular) or V
j<i (upper triangular) O The null matrix is defined by aij = 0 all i, j. 0 The identity matrix, I, is a diagonal matrix with all elements
=1. 0 A symmetric matrix A is square is such that aﬁ=aﬂ i,j=l,...,n. A. O A matrix is idempotent if AA 4. Properties of matrix operations.
Matrix addition and multiplication of conformable matrices: O Associative ll ABC AiBiC=(AiB)iC=Ai(BiC) (AB)C = A(BC) O Commutative AiBiCzBiCiA=AiCiB ABC at BCA at ACB 0 Identity element. A 9‘! o O
o D
AIzﬁ; A+N=0. c5912..“
(0
. . , (Lgxi) 3
O Distributive (—l)A + A = N; The inverse element for multiplication is
discussed in section 8. Econ 5000 II. Linear Algebra 5. Matrix Transposition Let A = {an} i=l,...,m;j=l,...,n be an mxn matrix. The transpose
of A is denoted by A? or A'. The transpose is defined as AF =
{aji} i=1, . ..,m;j=l, . ..,n. If A is an mxn matrix then A? is an nXm matrix. 9 (A+B+C)T = AT+BT+CT '1"!!! o (ABC)T= c B A
For a symmetric matrix, A = AF.
6. Determinants The determinant of a square matrix is a unique number associated
with that matrix. A determinant can be positive, negative or zero. 9 If A={aﬁ} i,j=l,...,n then the determinant of A (A)is defined as follows:
l'l
IAI: Elaﬁcﬁ where Cij is the cofactor of the i,jth element of A. C.. = (1)“j M.. 1] 1] where Mij is the minor associated with the i,jmelement of A. 9 MH is the determinant of the matrix that is formed when we
delete the ith row and jth column of A. EX‘ 3') A=[a11] : A[= an
a a
11 u
AL:
3 a
2) m 22
= = —1)2  + a (—l)3a  = a a —a a
: IAI a11C11+a12C12 a11( a2: 12 21 11 22 12 21 all 3'12 3'13
3 ) A = 32] a22 an
as] as: 333 =$A = a C + a C + a C 11 11 12 12 13 13 Econ 5000 II. Linear Algebra a a a a a a
2 22 23 3 21 23 4 21 22
=a11(_1) +a12(—1) +a13(—1)
332 333 331 333 a31 332
= a11(a22a33_a23a32) — a‘12 (azlan—azaaal) + a13(a21a12_a22a31) 7. Properties of Determinants O O O The value of the determinant is unchanged if corresponding rows
and columns are interchanged. Eg. [Al = AH If A is a square null matrix then A = O. The determinant of a triangular matrix is the product of the
diagonal elements. Since a diagonal matrix is a special case of
a triangular matrix, this is also true for diagonal matrices. If a new matrix is formed by nmltiplying every element in a
given row or column by a scalar (k) then the determinant of the
new matrix is k times the determinant of the old matrix. If a new matrix is formed by interchanging two rows or columns
of a given matrix then the determinant of the new matrix is (—1)
times that of the old matrix. If a new matrix is formed by adding to a given row (or column) a
linear combination of the other rows (or columns) in a matrix,
then the determinant of the new matrix is the same as that of
the old matrix. If two rows or columns in a matrix are identical, then the
determinant of the matrix is zero. In general, if any given row
or column of a matrix can be expressed as a linear combination
of some or all of the other rows or columns of the matrix, then
the determinant of that matrix is zero. For a matrix to have a
nonzero determinant it must be of full rank where the rank of a
matrix is the number of linearly independent rows (or columns)
in that matrix. IABI = A lBl In general A+B¢A+B 8. Matrix Inversion Let A be a square matrix of full rank (i.e. A¢O). Then there 6 Econ 5000 exists an inverse matrix, A“, such that: II. Linear Algebra IPA = I
The inverse is given by:
+
A“ A
W
where A‘ is the adjoint matrix. A‘ = CT where C = {cu}
i=1, .. .,n;j=1, . . .,n. C is the matrix of cofactors of A.
EX.
3 a
n 12
A:
321 a22
IA. _ a11a22 _ a12a21
C11 = (—1):a22
C12 : (_l)3 a‘21
C21 : (—1)4 an
C22 : (—1) an
Thus:
322 “321
C:
‘312 an
and
CT_ 322 _a12
—a21 an The inverse of A is therefore: A—1=_1_ “22 au IA! —a21 an ’ 9. Properties of Inverse Matrices 9 (11")'1 = A II. Linear Algebra Econ 5000
° IA'll = IAI'l = 1/A
. (AT)1. = (A1)T Econ 5000 II. Linear Algebra Systems of Linear Equations 10. Introductory Remarks. Consider the pair of equations: y=10~x
y=1+x
The solution to this system of equations is {x*,y*} = {4.5,S.5}. This system can also be written as: y+x=10
yx=1 [1 fiin] This is in the form Ax = b. If A4 exists we can write: Or alternatively: Kmx==x%::x==x% Note that we are assuming here that the matrix A and its inverse
are conformable with the vectors x and b as in the numerical
example above. Now the inverse of A is:
A_,_1 1"_.5 .5
‘1—1 “.5 —.5
We can use this to solve for x* and y*.
y*_.5 .510_5.5
x*'.5 —.5_1“4.5 Thus matrix inversion is a tool that we can use to solve a system
of linear equations. Econ 5000 II. Linear Algebra 1 1. Problem Situations Case 1: Consider the system of equations:
y=10—x
yzll—x A graph of these equations reveals that they are two parallel
lines and hence there is no solution to this system of equations. In matrix form: 11y_10
11x_11 A=O and hence A‘1 does not exist. Case 2: Consider the system of equations: y=10—x
.5y=5—.5x These equations represent the same line, the second equation is
just the first equation divided by two. More generally, the
equations are linearly dependent. Again there is no unique
solution to the system of equations — there are an infinite
number of solutions. In matrix form: 1 y
x
Ax=b
A— —O and hence the inverse of A does not exist. 12. General Theory of Linear Equations Defn: A set of m vectors aw i = 1,... ,m is linearly dependent if
there exists a set of m constants oi, i = 1,... ,m such that; m Elia! 2 i=1 and not all of the .i are equal to 0. Otherwise, the set of
vectors is said to be linearly independent. 10 Econ 5000 II. Linear Algebra Defn: For the general matrix A(an). The rank of A (r(A))is equal
to the number of linearly independent rows which is equivalent to
the number of linearly independent columns in A. r(A) S min[n.m] r(A) = the order of the largest square submatrix of A for which the
determinant is not zero. Remember, if a square matrix has one or
more linearly dependent rows then its determinant is zero. 9 If D is a triangular matrix then r(D) = the number of non—zero
diagonal elements. 0 r(AT) = r(A)
O r(AB) S min[r(A),r(B)]
o If A is an nxn matrix then r(A) = n iff ]A i O. 0 If A is square and A = 0, then A is singular. Let Ax = b be a system of equations in n unknowns. There are
several cases: 9 b = O (homogeneous case). Then if A i 0 then the unique
solution to this set of equations is x = O. O b i O (nonhomogeneous case). Define an augmented matrix by adjoining b to A as the (n+l)“'t column. Refer to this as the
partitioned matrix (Ab). 0 if r(Ab) = r(A) then the system of equations is consistent
and there exists at least one solution. 0 if r(Ab) = r(A) and r(A) = n then A i O and the unique
solution is given by x*=A4b. 9 if r(Ab) = r(A) and r(A) < n then A = O and there is an infinite number of solutions to the system. (Note that this
corresponds to the 2nd problem case). 0 If r(Ab) ¢ r(A) then the equations are inconsistent and no solution exists. (This corresponds to the first problem
case). 11 Econ 5000 II. Linear Algebra 13. Cramer’ Rule Consider a system of equations in n unknowns:
Ax=b;b¢0
Assume r(Ab) = r(A) = n. The system has a unique non trivial solution x* The ith element
of x*, x3} is given by: 39> * x.: ‘ IAI i is the matrix formed by replacing the ith column of A with b. 9 ex. (from introductory remarks);
10 1‘
1 —1 —11
x*=e————:———=55
‘ 1 1 —2
1 1 12 Econ 5000 II. Linear Algebra Quadratic Forms 14. Quadratic Form Quadratic form: the scalar which results when a given square
(an) matrix is pre and post multiplied by the same vector. ex. define: xl
x all ' am
1:2 2, A: , xT=[xl x2 xn]
an] "' arm
.X Q = xTAx = 22ayxixj i=l J=l 15. Positive and Negative Definite. Defn: If for all vectors x i 0, fix < 0, then A is said to be
negative definite (n.d.). Defn: If for all vectors x ¢ 0, xﬁb:>O, then A is said to be
positive definite (p.d.). 6 if A is n.d. , then A is p.d.; if A is p.d. ,then —A is n.d.. In economics almost all cases of negative and positive definiteness
arise when the matrix A is symmetric. In what follows we will
assume that A is symmetric. In the following we will examine the conditions on A in 2x2 and 3x3 cases and then deduce the general restrictions for p.d. and n.d.
quadratic forms. 13 Econ 5000 II. Linear Algebra A, symmetric 2x2 matrix: Conditions such that xTAx < 0 (>0) . 'r Q=xAx all all x1 _ 2 2
_ anxl + 20.1le.39 + xza'zz
(1 (122 X: =[x
' x 2
2 Completing the square in x1 we get: 2
a x 2 a a —a1 2 all all
Thus
0 Q < O ifza11 < 0 and ana11 — am2 > O.Note that all 2 a11and
(a22a'll—a12 ) = [Al ' O Q > 0 if an > O and A >0. 3x3 Case: an :112 a13 x1
_ T _
Q—x Ax—[x] x2 x3 a21 a22 2123 x2 a13 323 a3:! x3 +ax2+ax2+2ax +2axx+2axx
222 121% 3 2
—ax 333 131 2323 11 1
Completing the square yields:
an an a x a x a a
Q =la11(x1+—12 2 + '3 3)2 +——12 22 (x2 +
an an laul 2
a11a22 _ an 2
2 2 2 2
IA I x3 aha222133 _alla23 _ a225113 _ 21332112 +2a2a13a23 2
a11a22 —a12 +
an an a12 a2: 14 Econ 5000 II. Linear Algebra Defn: For a square nxn matrix A, the successive (leading) principal
minors of A are the determinants of the successively larger
submatrices formed by moving in a southeasterly direction along the
diagonal of A and taking these successive elements on the diagonal
as the (nxn)th element of the respective submatrices. EX .
an an an
A = am 322 a23 a31 a32 333
The leading principal minors of A are: an an FHL ’ IAI an an
Thus for Q = xhu: : A symmetric, O Q < 0 if the leading principal minors of A alternate in sign
with the first being negative. 0 Q > 0 if the leading principal minors of A are all positive. Note that in the case of either positive (or negative)
definiteness, the restriction that Ian] > 0 (< O) is a more general
restriction that aii > 0 (< 0) because we can always permute the
rows and columns of A without affecting its positive or negative
definiteness. 15 Econ 5000 II. Linear Algebra 16‘. Positive and Negative Definiteness Subject to Constraint In the foregoing discussion it was assumed that x was an arbitrary
(non—null) vector. In many circumstances, however, we may be
interested in only a restricted set of vectors 1: when attempting to
attach a general sign to Q = 1531:. In economics, for example, we
often are looking for the sign of Q when the 1: vector satisfy the restriction b”): = 0 where b is a real vector such that b :6 0.
Mia:
Q = aux: + 2aux1x2 + aux:
b’x = 0 ::> blx1 + b2}:2 = O ::> x2 = b1x1/b2
Substituting in 1:351: for x2 and collecting terms yields:
0 b1 b2 __ 2
—04)x1b an %2
b2 an an Q=xTAx bTx=0 The determinant appearing on the right hand side is the bordered
second minor of A. This must be positive for Q < O and negative for Q > 0.
3x3 Case: 0 b1 b2
<0 if bl all a12>0, b2 312 a22 1 all a12 a” Q TAxl b
:x T
bx=0
b2 a12 a22 a23.
b 3 a13 a:23. a33. In general (nxn): If the successive bordered principal minors of A alternate in
Sign starting with the second one > 0(as in the 2x2 case), then Q=xTAxI <0 bTx =0 If the successive bordered principal minors of A are all
negative, then Q=xTAx l >0 bTx=0
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 Summer '09
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