Econ 508-A

DEFINITE MATRICES IDENTIFYING DEFINITENESS AND SEMIDEFINITENESS DEFINITENESS ON SUBSPACES

Econ 508-ADefinite Matrices

Carmen Astorne-FigariWashington University in St. Louis

August 1, 2011


QUADRATIC FORMS

DEFINITION: A quadratic form on RN is a real valuedfunction of the form

Q(x1, ..., xN) =N∑

i=1

N∑j=1

aijxixj

where each term is a monomial of degree two.


MATRIX REPRESENTATION OF A QUADRATIC FORM

Let x = (x1, ..., xN).

Then Q(x) can also be represented in matrix form:

Q(x) = x′Ax

where A is a symmetric N ×N matrix.


EXAMPLE 1

The general quadratic form in R2 :

a11x21 + 2a12x1x2 + a22x2

2

can be written in matrix terms:

[x1 x2

] [a11 a12a12 a22

] [x1x2

]


EXAMPLE 2

The general quadratic form in R3 :

a11x21 + a22x2

2 + a33x23 + 2a12x1x2 + 2a13x1x3 + 2a23x2x3

can be written in matrix terms:

[x1 x2 x3

] a11 a12 a13a12 a22 a23a13 a23 a33

x1x2x3

And so forth


DEFINITE MATRICES

DEFINITION: Let A be a symmetric N ×N matrix. Then A is:

1. Positive Definite (PD) iff Q(x) = x′Ax > 0 ∀x ∈ RN\{0}

2. Positive Semidefinite (PSD) iff Q(x) = x′Ax ≥ 0 ∀x ∈ RN\{0}

3. Negative Definite (ND) iff Q(x) = x′Ax < 0 ∀x ∈ RN\{0}

4. Negative Semidefinite (NSD) iff Q(x) = x′Ax ≤ 0 ∀x ∈ RN\{0}


DEFINITE MATRICES (CONTINUED)

5. Indefinite if Q(x) = x′Ax > 0 for some x ∈ RN, and

Q(x) = x′Ax < 0 for some other x ∈ RN

REMARK: A matrix that is PD (ND) is automatically PSD(NSD). Otherwise, every symmetric matrix falls into one of thefive mentioned categories.


EXAMPLE 1

1. A =[

1 00 1

]= I

Pick any x ∈ R2\{0}

Q(x) =[x1 x2

] [1 00 1

] [x1x2

]= x2

1 + x22 > 0

So A is PD


EXAMPLE 3

3 C =[

0 11 0

]Pick any x ∈ R2\{0}

Q(x) = x′Cx = 2x1x2

xTCx = 2 > 0 for x = (1, 1)xTCx = −2 < 0 for x = (−1, 1)

So C is Indefinite


PICTURES

(i) Q(x) = x21 + x2

2

(ii) Q(x) = −x21 − x2

2

(iii) Q(x) = x21 − x2

2

(iv) Q(x) = x21 + 2x1x2 + x2

2

(v) Q(x) = −x21 − 2x1x2 − x2

2


IDENTIFYING DEFINITENESS AND SEMIDEFINITENESS

1. Eigenvalues

2. Principal minors


EIGENVALUES

Let A be a N ×N square matrix.

The following equation is called characteristic equation:

det[A− λI] = 0

The solutions to the characteristic equation are calledcharacteristic roots or eigenvalues.


EXAMPLE

A =[

2 −6−6 −7

]The characteristic equation is:

det[A− λI] = 0∣∣∣∣[ 2 −6−6 −7

]−[λ 00 λ

]∣∣∣∣ = 0

∣∣∣∣[2− λ −6−6 −7− λ

]∣∣∣∣ = 0


EXAMPLE 1 (CONTINUED)

(2− λ)(−7− λ)− (−6)(−6) = 0

λ2 + 5λ− 50 = 0

(λ− 5)(λ+ 10) = 0

λ1 = 5

λ2 = −10


IDENTIFYING DEFINITENESS USING EIGENVALUES

THEOREM: Let A be an N ×N symmetric matrix. Then

1. A is PD iff all its eigenvalues are positive.

2. A is PSD iff all its eigenvalues are nonnegative.

3. A is ND iff all its eigenvalues are negative.

4. A is NSD iff all its eigenvalues are nonpositive.

5. A is indefinite iff it has at least one positive eigenvalue and atleast one negative eigenvalue.


EXAMPLE 1

1. A =[

1 00 1

]The characteristic equation is

|I − λI| =∣∣∣∣[1− λ 0

0 1− λ

]∣∣∣∣ = (1− λ)2

λ1 = λ2 = 1

So A is PD


EXAMPLE 2

2. B =[

1 00 0

]The characteristic equation is∣∣∣∣[1 0

0 0

]− λI

∣∣∣∣ = ∣∣∣∣[1− λ 00 −λ

]∣∣∣∣ = (1− λ)(−λ) = 0

λ1 = 0λ2 = 1

So B is PSD


EXAMPLE 3

3. C =[

0 11 0

]The characteristic equation is∣∣∣∣[0 1

1 0

]− λI

∣∣∣∣ = ∣∣∣∣[−λ 11 −λ

]∣∣∣∣ = (−λ)2 − 1 = 0

λ1 = −1λ2 = 1

So C is Indefinite


PRINCIPAL MINORS

DEFINITION: Let A be an N ×N square matrix.

1. The K × K submatrix obtained from A by deleting any (N − K)columns of A and the corresponding (N − K) rows of A is calledK-order principal submatrix of A.

2. The determinant of a K-order principal submatrix of A is called aK-order principal minor (principal minor (PM) of order K).

3. The K-order principal submatrix of A obtained by deleting the last(N − K) columns of A and the last (N − K) rows of A is called theK-order leading principal submatrix of A, denoted AK.(An N ×N matrix has N leading principal submatrices.)

4. The determinant of AK is called the K-order leading principalminor (LPM), denoted |AK|.


EXAMPLE

A =

a11 a12 a13a12 a22 a23a13 a23 a33

Leading principal submatrices and leading principal minors:

A1 =[a11]

; |A1| = a11

A2 =[

a11 a12a21 a22

]; |A2| = a11a22 − a12a21

A3 =

a11 a12 a13a21 a22 a23a31 a32 a33

; |A|


EXAMPLE (CONTINUED)

Principal submatrices and principal minors:

Of order 1:[a22]

a22[a33]

a33

Of order 2:[a11 a13a31 a33

] ∣∣∣∣[a11 a13a31 a33

]∣∣∣∣ = a11a33 − a13a31

[a22 a23a32 a33

] ∣∣∣∣[a22 a23a32 a33

]∣∣∣∣ = a22a33 − a23a32


IDENTIFYING DEFINITENESS USING PRINCIPAL

MINORS

THEOREM: Let A be an N ×N symmetric matrix. Then

1. A is PD iff all its LPM’s are positive.

2. A is PSD iff all its PM’s are nonnegative.

3. A is ND iff its LPM’s alternate in signs with (−1)K |AK| > 0 :every LPM of odd order is negative

every LPM of even order is positive

4. A is NSD iff every PM of odd order is nonpositiveevery PM of even order is nonnegative.

5. A is indefinite iff some LPM’s of order k are nonzero but do notfit into (1) or (3).


EXAMPLE 1

1. A =[

1 00 1

]Look at the AK’s and LPM’s first:

A1 =[1], |A1| = 1

A2 =[

1 00 1

]|A2| = 1

So A is PD


EXAMPLE 2

2 B =[

1 00 0

]Look at the BK’s and LPM’s first:

B1 =[1], |B1| = 1

B2 =[

1 00 0

]|B2| = 0

So we know that B is not PD. Is it PSD?

Check all the remaining principal submatrices and PM’s

Of order 1:[0] ∣∣[0]∣∣ = 0

So B is PSD


EXAMPLE 3

3. C =[

0 11 0

]Look at the CK’s and LPM’s first:

C1 =[0], |C1| = 0

C2 =[

0 11 0

]|C2| = −1

Since the second order LPM is negative, C doesn’t fit into anycategory

So C is Indefinite


DEFINITENESS ON SUBSPACES

Let B be a N ×M matrix, A be N ×N, x be N × 1.

The set

T = {x ∈ RN : B′x = 0 }

consists of all vectors that are orthogonal to the columns of B.

T is a vector subspace in RN.

If the columns of B are linearly independent, T is N −Mdimensional.


DEFINITE QUADRATIC FORMS

THEOREM: Let A be symmetric, B of rank M. The quadraticform x′Ax > 0 ∀ x ∈ T (is PD on T) iff

(−1)M∣∣∣∣ARR BRM

B′

MR0

∣∣∣∣ > 0 for R = M + 1, ...,N.

That is, the border preserving LPM’s of orders M + 1, ...,Nhave the same sign as (−1)M .


DEFINITE QUADRATIC FORMS (CONTINUED)

THEOREM: Let A be symmetric, B of rank M. The quadraticform x′Ax < 0 ∀ x ∈ T (is ND on T) iff

(−1)R∣∣∣∣ARR BRM

B′

MR0

∣∣∣∣ > 0 for R = M + 1, ...,N.

In this case, the border preserving LPM’s of ordersM + 1, ...,N have the same sign as (−1)R forR = M + 1, ...,N. That is, they alternate signs.


EXAMPLE

Let A =

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34a41 a42 a43 a44

B =

b11 b12b21 b22b31 b32b41 b42

So N = 4, M = 2, R = 3, 4

The bordered matrix is A =

a11 a12 a13 a14 b11 b12a21 a22 a23 a24 b21 b22a31 a32 a33 a34 b31 b32a41 a42 a43 a44 b41 b42b11 b21 b31 b41 0 0b12 b22 b23 b24 0 0

The border preserving leading principal submatrix of order R isobtained by deleting the rows and columns of A corresponding to thlast N − R rows and columns of the original matrix.


EXAMPLE (CONTINUED)

The border preserving leading principal submatrix of order 3 is:

a11 a12 a13 b11 b12a21 a22 a23 b21 b22a31 a32 a33 b31 b32b11 b21 b31 0 0b12 b22 b23 0 0

For semidefiniteness, the respective requierments extend to allthe border preserving principal minors. Also, the inequalityregarding the sign is weak.

Econ 508-A

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