Linear Algebra And Its Applications Chapter 6. Positive Definite Matrix

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1 Linear Algebra And Its Applications Chapter 6. Positive Definite Matrix KAIST wit Lab 남성호

2 Introduction The signs of the eigenvalues can be important. The signs can also be related to the minima, maxima, and saddle points. The new and highly important problem is to recognize a minimum point. Check the second derivative test 0 into n dimensions. Example) Does either, or, have a minimum at point at 0? 2

3 Example The first derivatives at (0,0),, is a stationary point for both functions. The second derivatives at (0,0) The two functions behave in exactly the same way near the origin. has a minimum if and only if has a minimum. 3

4 Quadratic form Quadratic form Def) In mathematics, Quadratic form is a homogenous polynomial of degree two in a number of variables. Every quadratic form has a stationary point at the origin. A quadratic can be expressed by a symmetric matrix. Example) If the stationary point of is at,, 4

5 Positive/Negative (Semi-)Definite A quadratic form is positive/negative definite if the values are positive/negative for all points except for the origin. A quadratic form is positive/negative semidefinite if the values are non negative/non positive. If is positive definite, has a minimum Example) 5

6 Condition for Positive Definite Example), 2 0, : Positive definite has a minimum. 0, : Negative definite has a maximum. ( has minimum.) : Positive semidefinte ( 0 : Negative semidefinite ( 0) : indefinite has a saddle point 6

7 Positive definite matrix The positive definiteness of a matrix is defined via Positive definite matrix Def) An n by n symmetric matrix is positive definite matrix if and only if 0 for all non zero vectors. 0 the matrix is positive definite Example) In two dimensional case is positive definite iff 0and 7

8 Problem 8

9 Tests for Positive Definiteness Each of the following tests is a necessary and sufficient condition for the real symmetric matrix to be positive definite: 1) 0 for all nonzero real vectors 2) All the eigenvalues of satisfy 0 3) All the upper left submatrices have positive determinants 4) All the pivots (without row exchanges) satisfy 0 5) There is a matrix with independent columns such that 9

10 Proof 1) Definition 1) 2) 2) 1) 10

11 Proof 1) 3) The determinant is the product of the eigenvalues. If 1) holds, we already know that these eigenvalues are positive. 3) 4) d k det det A k Ak 1 0 4) 1) From Gaussian Elimination, 11

12 1) 5) Proof Suppose, then. If R has independent columns, 0 (except 0). There is a matrix with independent columns such that 12

13 Tests for Positive Semnidefiniteness Each of the following tests is a necessary and sufficient condition for the real symmetric matrix to be positive semidefinite: I. 0 for all nonzero real vectors. II. All the eigenvalues of satisfy 0. III. No principal submatrices have negative determinants. IV. No pivots are negative. V. There is a matrix, possibly with dependent columns, such that. 13

14 Ellipsoid If is positive definite, 1 is the equation of an ellipsoid. The axes of the ellipsoid point toward the eigenvector of Its axes have lengths,,, from the center. Example) 14

15 Congruence transformation The Law of Inertia Congruence transform is related to quadratic form The signs of the eigenvalues are preserved by a congruence transformation. has the same number of positive eigenvalues, negative eigenvalues and zero eigenvalues as. For any symmetric matrix, the signs of the pivots agree with the signs of the eigenvalues. The eigenvalue matrix Λ and the pivot matrix have the same number of positive entries, negative entries, and zero entries. 15

16 The Generalized Eigenvalue Problem Example) 16

17 Equivalent Eigenvalue Problem. ( is assumed to be positive definite) Let. Then Let. Then The eigenvalues are the same as for the original, and the eigenvectors are related by Property (when is symmetric, is positive definite) The eigenvalues for are real. The s have same signs as the eigenvalues of. Chas orthogonal eigenvectors. So the eigenvectors of have 17

18 Problem 18

19 Problem 19

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