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EC400 Part II, Math for Micro: Lecture 1
Leonardo Felli
NAB.SZT
9 September 2010
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Course Outline
Lecture 1: Tools for optimization (Quadratic forms).
Lecture 2: Tools for optimization (Taylor’s expansion) andUnconstrained optimization.
Lecture 3: Concavity, convexity, quasi-concavity and economicapplications.
Lecture 4: Constrained Optimization I: Equality Constraints,Lagrange Theorem.
Lecture 5: Constrained Optimization II: Inequality Constraints,Kuhn-Tucker Theorem.
Lecture 6: Constrained Optimization III: The Maximum ValueFunction, Envelope Theorem, Implicit Function Theorem andComparative Statics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 2 / 27
Admin
My coordinates: S.478, x7525, [email protected]
PA: Gill Wedlake, S.379, x6889, [email protected]
Office Hours:
Thursday 9 September — 10:00-12:00 a.m.Friday 10 September — 10:00-12:00 p.m.Monday 13 September — 10:00-12:00 a.m.Tuesday 14 September — 10:00-12:00 a.m.Wednesday 15 September — 10:00-12:00 a.m.Thursday 16 September — 10:00-12:00 a.m.Friday 17 September — 10:00-12:00 a.m.Wednesday 22 September — 10:00-12:00 a.m.
or by appointment (e-mail [email protected]).
Course Material: available at:http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27
Admin
My coordinates: S.478, x7525, [email protected]
PA: Gill Wedlake, S.379, x6889, [email protected]
Office Hours:
Thursday 9 September — 10:00-12:00 a.m.Friday 10 September — 10:00-12:00 p.m.Monday 13 September — 10:00-12:00 a.m.Tuesday 14 September — 10:00-12:00 a.m.Wednesday 15 September — 10:00-12:00 a.m.Thursday 16 September — 10:00-12:00 a.m.Friday 17 September — 10:00-12:00 a.m.Wednesday 22 September — 10:00-12:00 a.m.
or by appointment (e-mail [email protected]).
Course Material: available at:http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27
Admin
My coordinates: S.478, x7525, [email protected]
PA: Gill Wedlake, S.379, x6889, [email protected]
Office Hours:
Thursday 9 September — 10:00-12:00 a.m.Friday 10 September — 10:00-12:00 p.m.Monday 13 September — 10:00-12:00 a.m.Tuesday 14 September — 10:00-12:00 a.m.Wednesday 15 September — 10:00-12:00 a.m.Thursday 16 September — 10:00-12:00 a.m.Friday 17 September — 10:00-12:00 a.m.Wednesday 22 September — 10:00-12:00 a.m.
or by appointment (e-mail [email protected]).
Course Material: available at:http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27
Admin
My coordinates: S.478, x7525, [email protected]
PA: Gill Wedlake, S.379, x6889, [email protected]
Office Hours:
Thursday 9 September — 10:00-12:00 a.m.Friday 10 September — 10:00-12:00 p.m.Monday 13 September — 10:00-12:00 a.m.Tuesday 14 September — 10:00-12:00 a.m.Wednesday 15 September — 10:00-12:00 a.m.Thursday 16 September — 10:00-12:00 a.m.Friday 17 September — 10:00-12:00 a.m.Wednesday 22 September — 10:00-12:00 a.m.
or by appointment (e-mail [email protected]).
Course Material: available at:http://econ.lse.ac.uk/staff/lfelli/teaching and Moodle
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 3 / 27
Suggested Textbooks
Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther Mathematics for Economic Analysis.
Alpha C. Chiang Fundamental Methods of Mathematical Economics.
Carl P. Simon and Lawrence E. Blume Mathematics for Economists.
Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization:The Calculus of Variations and Optimal Control in Economics andManagement.
Akira Takayama Mathematical Economics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27
Suggested Textbooks
Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther Mathematics for Economic Analysis.
Alpha C. Chiang Fundamental Methods of Mathematical Economics.
Carl P. Simon and Lawrence E. Blume Mathematics for Economists.
Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization:The Calculus of Variations and Optimal Control in Economics andManagement.
Akira Takayama Mathematical Economics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27
Suggested Textbooks
Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther Mathematics for Economic Analysis.
Alpha C. Chiang Fundamental Methods of Mathematical Economics.
Carl P. Simon and Lawrence E. Blume Mathematics for Economists.
Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization:The Calculus of Variations and Optimal Control in Economics andManagement.
Akira Takayama Mathematical Economics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27
Suggested Textbooks
Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther Mathematics for Economic Analysis.
Alpha C. Chiang Fundamental Methods of Mathematical Economics.
Carl P. Simon and Lawrence E. Blume Mathematics for Economists.
Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization:The Calculus of Variations and Optimal Control in Economics andManagement.
Akira Takayama Mathematical Economics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27
Suggested Textbooks
Knut Sydsaeter, Peter Hammond, Atle Seierstad and Arne StromFurther Mathematics for Economic Analysis.
Alpha C. Chiang Fundamental Methods of Mathematical Economics.
Carl P. Simon and Lawrence E. Blume Mathematics for Economists.
Morton I. Kamien and Nancy L. Schwartz Dynamic Optimization:The Calculus of Variations and Optimal Control in Economics andManagement.
Akira Takayama Mathematical Economics.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 4 / 27
What is a quadratic form?
Quadratic forms are useful because:
(i) The simplest functions after linear ones;
(ii) Conditions for optimization techniques are stated in terms ofquadratic forms;
(iii) Economic optimization problems have a quadratic objective function,such as risk minimization problems in finance, where riskiness ismeasured by the quadratic variance of the returns from investments.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27
What is a quadratic form?
Quadratic forms are useful because:
(i) The simplest functions after linear ones;
(ii) Conditions for optimization techniques are stated in terms ofquadratic forms;
(iii) Economic optimization problems have a quadratic objective function,such as risk minimization problems in finance, where riskiness ismeasured by the quadratic variance of the returns from investments.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27
What is a quadratic form?
Quadratic forms are useful because:
(i) The simplest functions after linear ones;
(ii) Conditions for optimization techniques are stated in terms ofquadratic forms;
(iii) Economic optimization problems have a quadratic objective function,such as risk minimization problems in finance, where riskiness ismeasured by the quadratic variance of the returns from investments.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 5 / 27
Among the functions of one variable, the simplest functions with aunique global extremum are the quadratic forms:
y = x2 and y = −x2.
The level curve of a general quadratic form in R2 is
a11x21 + a12x1x2 + a22x
22 = b
and can take the form of an ellipse, a hyperbola, a pair of lines, apoint, or possibly, the empty set.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 6 / 27
Among the functions of one variable, the simplest functions with aunique global extremum are the quadratic forms:
y = x2 and y = −x2.
The level curve of a general quadratic form in R2 is
a11x21 + a12x1x2 + a22x
22 = b
and can take the form of an ellipse, a hyperbola, a pair of lines, apoint, or possibly, the empty set.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 6 / 27
Definition of a Quadratic Form
Definition
Definition: A quadratic form on Rn is a real valued function
Q(x1, x2, ..., xn) =∑i≤j
aijxixj
The general quadratic form of
a11x21 + a12x1x2 + a22x
22
can be written (non uniquely) as:
(x1 x2
) ( a11 a120 a22
) (x1x2
)
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 7 / 27
Definition of a Quadratic Form
Definition
Definition: A quadratic form on Rn is a real valued function
Q(x1, x2, ..., xn) =∑i≤j
aijxixj
The general quadratic form of
a11x21 + a12x1x2 + a22x
22
can be written (non uniquely) as:
(x1 x2
) ( a11 a120 a22
) (x1x2
)
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 7 / 27
Each quadratic form can be represented as
Q(x) = xTA x
where A is a (unique) symmetric matrix:a11 a12/2 ... a1n/2a12/2 a22 ... a2n/2
......
. . ....
a1n/2 a2n/2 ... ann
Conversely, if A is a symmetric matrix, then the real valued functionQ(x) = xTA x, is a quadratic form.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 8 / 27
Each quadratic form can be represented as
Q(x) = xTA x
where A is a (unique) symmetric matrix:a11 a12/2 ... a1n/2a12/2 a22 ... a2n/2
......
. . ....
a1n/2 a2n/2 ... ann
Conversely, if A is a symmetric matrix, then the real valued functionQ(x) = xTA x, is a quadratic form.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 8 / 27
Definiteness of Quadratic Forms
A quadratic form always takes the value 0 when x = 0.
We focus on the question of whether x = 0 is a max, a min, orneither.
Consider for example:y = ax2
then if a > 0, ax2 is non negative and equals 0 only when x = 0. Thisis positive definite, and x = 0 is a global minimizer.
If a < 0, then the function is negative definite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 9 / 27
Definiteness of Quadratic Forms
A quadratic form always takes the value 0 when x = 0.
We focus on the question of whether x = 0 is a max, a min, orneither.
Consider for example:y = ax2
then if a > 0, ax2 is non negative and equals 0 only when x = 0. Thisis positive definite, and x = 0 is a global minimizer.
If a < 0, then the function is negative definite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 9 / 27
In two dimensions, for example:
x21 + x22
is positive definite, whereas
−x21 − x22
is negative definite, whereas
x21 − x22
is indefinite, since it can take both positive and negative values.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 10 / 27
Semi-definiteness of Quadratic Forms
There are two “intermediate” cases:
If the quadratic form is always non negative but also equals 0 for nonzero x′s, is called positive semi-definite, for example:
(x1 + x2)2
which can be 0 for points such that x1 = −x2.
A quadratic form which is never positive but can be zero at pointsother than the origin is called negative semidefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 11 / 27
Semi-definiteness of Quadratic Forms
There are two “intermediate” cases:
If the quadratic form is always non negative but also equals 0 for nonzero x′s, is called positive semi-definite, for example:
(x1 + x2)2
which can be 0 for points such that x1 = −x2.
A quadratic form which is never positive but can be zero at pointsother than the origin is called negative semidefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 11 / 27
Definiteness and Semi-definiteness of Symmetric Matrixes
We apply the same terminology to the symmetric matrix A, such that:
Q(x) = xTA x
Definition
Let A be an (n × n) symmetric matrix. Then A is:
positive definite if xTA x > 0 for all x 6= 0 in Rn,
positive semi-definite if xTA x ≥ 0 for all x 6= 0 in Rn,
negative definite if xTA x < 0 for all x 6= 0 in Rn,
negative semi-definite if xTA x ≤ 0 for all x 6= 0 in Rn,
indefinite xTA x > 0 for some x 6= 0 in Rn and xTA x < 0 for somex 6= 0 in Rn.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 12 / 27
Definiteness and Semi-definiteness of Symmetric Matrixes
We apply the same terminology to the symmetric matrix A, such that:
Q(x) = xTA x
Definition
Let A be an (n × n) symmetric matrix. Then A is:
positive definite if xTA x > 0 for all x 6= 0 in Rn,
positive semi-definite if xTA x ≥ 0 for all x 6= 0 in Rn,
negative definite if xTA x < 0 for all x 6= 0 in Rn,
negative semi-definite if xTA x ≤ 0 for all x 6= 0 in Rn,
indefinite xTA x > 0 for some x 6= 0 in Rn and xTA x < 0 for somex 6= 0 in Rn.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 12 / 27
Application (later this week)
A function y = f (x) of one variable is concave on some interval if itssecond derivative f ′′(x) ≤ 0 on that interval.
The generalization of this result to higher dimensions states that afunction is concave on some region if its matrix of second derivatives(Hessian matrix) is negative semi-definite for all x in the region.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 13 / 27
Application (later this week)
A function y = f (x) of one variable is concave on some interval if itssecond derivative f ′′(x) ≤ 0 on that interval.
The generalization of this result to higher dimensions states that afunction is concave on some region if its matrix of second derivatives(Hessian matrix) is negative semi-definite for all x in the region.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 13 / 27
The Determinant
Definition
The determinant of a matrix is a unique scalar associated with the matrix.
The determinant of a (2× 2) matrix A =
(a11 a12a21 a22
)is:
det(A) = |A| = a11a22 − a12a21
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 14 / 27
The Determinant
Definition
The determinant of a matrix is a unique scalar associated with the matrix.
The determinant of a (2× 2) matrix A =
(a11 a12a21 a22
)is:
det(A) = |A| = a11a22 − a12a21
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 14 / 27
The determinant of a (3× 3) matrix A =
a11 a12 a13a21 a22 a23a31 a32 a33
:
det(A) = a11 det
(a22 a23a32 a33
)− a12 det
(a21 a23a31 a33
)+ a13 det
(a21 a22a31 a32
).
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 15 / 27
Definition
Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deletingn − k columns, say columns i1, i2, ..., in−k and the same n − k rows fromA, i1, i2, ..., in−k , is called a kth order principal submatrix of A.
Definition
The determinant of a k × k principal submatrix is called a kth orderprincipal minor of A.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 16 / 27
Definition
Let A be an (n × n) matrix. A (k × k) submatrix of A formed by deletingn − k columns, say columns i1, i2, ..., in−k and the same n − k rows fromA, i1, i2, ..., in−k , is called a kth order principal submatrix of A.
Definition
The determinant of a k × k principal submatrix is called a kth orderprincipal minor of A.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 16 / 27
Example
Consider a general (3× 3) matrix A.
There is one third order principal minor: det(A).
There are three second ordered principal minors:∣∣∣∣ a11 a12a21 a22
∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33
∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33
∣∣∣∣There are three first order principal minors: a11, a22 and a33.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27
Example
Consider a general (3× 3) matrix A.
There is one third order principal minor: det(A).
There are three second ordered principal minors:∣∣∣∣ a11 a12a21 a22
∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33
∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33
∣∣∣∣There are three first order principal minors: a11, a22 and a33.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27
Example
Consider a general (3× 3) matrix A.
There is one third order principal minor: det(A).
There are three second ordered principal minors:∣∣∣∣ a11 a12a21 a22
∣∣∣∣ , ∣∣∣∣ a11 a13a31 a33
∣∣∣∣ , ∣∣∣∣ a22 a23a32 a33
∣∣∣∣There are three first order principal minors: a11, a22 and a33.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 17 / 27
Definition
Let A be an (n × n) matrix. The kth order principal submatrix of Aobtained by deleting the last n − k rows and columns from A is called thekth order leading principal submatrix of A denoted Ak .
Definition
The determinant of the kth order leading principal submatrix is called thekth order leading principal minor of A denoted |Ak |.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 18 / 27
Definition
Let A be an (n × n) matrix. The kth order principal submatrix of Aobtained by deleting the last n − k rows and columns from A is called thekth order leading principal submatrix of A denoted Ak .
Definition
The determinant of the kth order leading principal submatrix is called thekth order leading principal minor of A denoted |Ak |.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 18 / 27
Testing Definiteness
Let A be an (n × n) symmetric matrix. Then:
A is positive definite if and only if all its n leading principal minors arestrictly positive.
A is negative definite if and only if all its n leading principal minorsalternate in sign as follows:
|A1| < 0, |A2| > 0, |A3| < 0 . . .
The kth order leading principal minor should have the same sign of(−1)k .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 19 / 27
Testing Definiteness
Let A be an (n × n) symmetric matrix. Then:
A is positive definite if and only if all its n leading principal minors arestrictly positive.
A is negative definite if and only if all its n leading principal minorsalternate in sign as follows:
|A1| < 0, |A2| > 0, |A3| < 0 . . .
The kth order leading principal minor should have the same sign of(−1)k .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 19 / 27
A is positive semi-definite if and only if every principal minor of A isnon negative.
A is negative semi-definite if and only if every principal minor of oddorder is non positive and every principal minor of even order is nonnegative:
|A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 20 / 27
A is positive semi-definite if and only if every principal minor of A isnon negative.
A is negative semi-definite if and only if every principal minor of oddorder is non positive and every principal minor of even order is nonnegative:
|A1| ≤ 0, |A2| ≥ 0, |A3| ≤ 0 . . .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 20 / 27
Special Case: Diagonal Matrixes
Consider the following (3× 3) matrix A, a diagonal one.
A =
a1 0 00 a2 00 0 a3
A also corresponds to the simplest quadratic forms:
a1x21 + a2x
22 + a3x
23 .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 21 / 27
Special Case: Diagonal Matrixes
Consider the following (3× 3) matrix A, a diagonal one.
A =
a1 0 00 a2 00 0 a3
A also corresponds to the simplest quadratic forms:
a1x21 + a2x
22 + a3x
23 .
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 21 / 27
This quadratic form will then be:
positive (negative) definite if and only if all the ai s are positive(negative).
It will be positive semi-definite if and only if all the ai s are nonnegative and negative semi-definite if and only if all the ai s are nonpositive.
If there are two ai s of opposite sign, it will be indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27
This quadratic form will then be:
positive (negative) definite if and only if all the ai s are positive(negative).
It will be positive semi-definite if and only if all the ai s are nonnegative and negative semi-definite if and only if all the ai s are nonpositive.
If there are two ai s of opposite sign, it will be indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27
This quadratic form will then be:
positive (negative) definite if and only if all the ai s are positive(negative).
It will be positive semi-definite if and only if all the ai s are nonnegative and negative semi-definite if and only if all the ai s are nonpositive.
If there are two ai s of opposite sign, it will be indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 22 / 27
Special Case: (2× 2) Matrixes
Consider the (2× 2) symmetric matrix:
A =
(a bb c
)
The associated quadratic form is:
Q(x1, x2) = xTA x = (x1, x2)
(a bb c
)(x1x2
)=
ax21 + 2bx1x2 + cx22
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 23 / 27
Special Case: (2× 2) Matrixes
Consider the (2× 2) symmetric matrix:
A =
(a bb c
)
The associated quadratic form is:
Q(x1, x2) = xTA x = (x1, x2)
(a bb c
)(x1x2
)=
ax21 + 2bx1x2 + cx22
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 23 / 27
If a = 0, then Q cannot be negative or positive definite sinceQ(1, 0) = 0.
Assume that a 6= 0 and add and subtract b2x22/a to get:
Q(x1, x2) = ax21 + 2bx1x2 + cx22 +b2
ax22 −
b2
ax22
= a(x21 +2bx1x2
a+
b2
a2x22 )− b2
ax22 + cx22
= a(x1 +b
ax2)2 +
(ac − b2)
ax22
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 24 / 27
If a = 0, then Q cannot be negative or positive definite sinceQ(1, 0) = 0.
Assume that a 6= 0 and add and subtract b2x22/a to get:
Q(x1, x2) = ax21 + 2bx1x2 + cx22 +b2
ax22 −
b2
ax22
= a(x21 +2bx1x2
a+
b2
a2x22 )− b2
ax22 + cx22
= a(x1 +b
ax2)2 +
(ac − b2)
ax22
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 24 / 27
If both coefficients, a and (ac − b2)/a are positive, then Q will neverbe negative.
It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0and x2 = 0. In other words, if
|a| > 0 and det(A) =
∣∣∣∣ a bb c
∣∣∣∣ > 0
then Q(x1, x2) is positive definite.
Conversely, in order for Q to be positive definite, we need both a anddet(A) =
(ac − b2
)to be positive.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27
If both coefficients, a and (ac − b2)/a are positive, then Q will neverbe negative.
It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0and x2 = 0. In other words, if
|a| > 0 and det(A) =
∣∣∣∣ a bb c
∣∣∣∣ > 0
then Q(x1, x2) is positive definite.
Conversely, in order for Q to be positive definite, we need both a anddet(A) =
(ac − b2
)to be positive.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27
If both coefficients, a and (ac − b2)/a are positive, then Q will neverbe negative.
It will equal 0 only when x1 + b x2/a = 0 and x2 = 0 or when x1 = 0and x2 = 0. In other words, if
|a| > 0 and det(A) =
∣∣∣∣ a bb c
∣∣∣∣ > 0
then Q(x1, x2) is positive definite.
Conversely, in order for Q to be positive definite, we need both a anddet(A) =
(ac − b2
)to be positive.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 25 / 27
Similarly, Q will be negative definite if and only if both coefficientsare negative, which occurs if and only if
a < 0 and(ac − b2
)> 0.
That is, when the leading principal minors alternative in sign.
If(ac − b2
)< 0. then the two coefficients will have opposite signs
and Q(x1, x2) will be indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 26 / 27
Similarly, Q will be negative definite if and only if both coefficientsare negative, which occurs if and only if
a < 0 and(ac − b2
)> 0.
That is, when the leading principal minors alternative in sign.
If(ac − b2
)< 0. then the two coefficients will have opposite signs
and Q(x1, x2) will be indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 26 / 27
Examples of a (2× 2) matrixes
Consider A =
(2 33 7
). Since |A1| = 2 and |A2| = 5, A is positive
definite.
Consider B =
(2 44 7
). Since |B1| = 2 and |B2| = −2, B is
indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 27 / 27
Examples of a (2× 2) matrixes
Consider A =
(2 33 7
). Since |A1| = 2 and |A2| = 5, A is positive
definite.
Consider B =
(2 44 7
). Since |B1| = 2 and |B2| = −2, B is
indefinite.
Leonardo Felli (LSE, NAB.SZT) EC400 Part II, Math for Micro: Lecture 1 9 September 2010 27 / 27