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TERM PAPER

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MTH-101

TOPIC: Real quadratic form Q= CX & it’s symmetric matrix c=[ ]

SUBMITTED TO: SUBMITTED BY:

Mr.KRISHNA PRASAD

NAME: ABHISHEK SINGH

ROLL NO.:B-39

SECTION:RE1906

REG : 10907589

ACKNOWLEDGEMENT

I would like to show my gratitude towards the help and guidance rendered to me by my teacher Mr. Krishna Prasad. I would like to acknowledge the support and help of lovely wi-fi network by which I received information of this term paper and I also thanks my family & friends, for completing this term paper.

ABHISHEK SINGH

Abstract:

A quadratic positive definite functional that yields necessary and sufficient conditions for the asymptotic stability of the solutions of the matrix difference-differential equation

is constructed and its structure is analyzed. This functional, a Liapunov functional, provides the best possible estimate for the rates of growth or decay of the solutions of this equation. The functional obtained, and its method of construction, are natural generalizations of the same problem for ordinary differential equations, and this relationship is emphasized. An example illustrates the applicability of the results obtained.

Contents: Introduction Explaination Diagonalation of symmetric matrix Theorem Question References

INTRODUCTION:a function f : Rn → R of the form

f(x) = Ax = Aijxixj

is called a quadratic form

in a quadratic form we may as well assume A = since

Ax = ((A + )/2)x

((A + )/2 is called the symmetric part of A)

uniqueness: if Ax = Bx for all x ∈ Rn and A = , B = , then

A = B

Explaination:

Let M be a symmetric n×n matrix, and Q(x) = Mx its associated quadratic form.

We wantmethods to tell whether M is positive definite, that is, whether Q(x) > 0 for every x 6= 0.For simplicity, we will assume that M is invertible, so that zero is not an eigenvalue.

1. Necessary conditions

1. In order for M to be positive definite, the diagonal elements must all be positive. Likewise, for M to be negative definite, the diagonal elements must all be negative.

2. The determinant of M is the product of its eigenvalues.

If the size n is even and det(M) < 0, then the eigenvalues must have different signs, so M is neither positive nor negative definite. If n is odd and the determinant is negative, the eigenvalues are not all positive. If n is odd and the determinant is positive, the eigenvalues are not all negative.

2. Necessary and sufficient conditions

It is always possible, and sometimes easy, to write Q(x) as a sum or difference of squares, which makes positive or negative definiteness transparent.

Let p(¸) = det(M − ¸I) denote the characteristic polynomial of M. If there is a computer or graphing calculator handy we can approximate the eigenvalues - the rootsofp(¸)numerically easily enough. But there is an easier way to determine positive definiteness. Because there are no non-real (complex) roots of p(¸), Descartes’ Rule of Signs says that the number of positive roots of p(¸) is the number of changes of sign in its coefficients.

Thus M is positive definite if and only if p(¸) has n changes of sign in its coefficients. Likewise, M is negative definite if and only if p(¸) has no changes of sign in its coefficients. Here is a method that avoids computing p(¸).

If the diagonal of M is all positive, then M is positive definite if and only if the determinants of all the upper left-hand corners are positive.

If the diagonal of M is all negative, then M is negative definite if and only if the determinantsof all the upper left-hand corners are alternate in sign.

Diagonalization of Symmetric Matrices:

A symmetric matrix is a square matrix such that = A. A matrix A is said to be

orthogonally

diagonalizable if there are an orthogonal matrix P (so ) and a diagonal matrix

D such

that A = PD = PD . An orthogonally diagonalizable matrix A with orthonormal

eigenvectors

. . . can be written as A = +. . .+ . This representation of A is

called a spectraldecomposition of A. Furthermore, each matrix ujujT is a projection matrix.1.2 Quadratic FormsA quadratic form on Rn is a function Q defined on Rn whose value at a vector x in Rn can be computed

by an expression of the form Q(x) = Ax, where A is an n × n symmetric matrix. The

matrix A iscalled the matrix of the quadratic form.If x represents a variable vector in Rn, then a change of variable is an equation of the

form x = ,

where P is an invertible matrix and y is a new variable vector in Rn. Now xTAx = yT (

AP)y. If P

diagonalizes A, then AP = AP = D, in which D is a diagonal matrix. The

columns op P arecalled the principal axes of the quadratic form xTAx.A quadratic form Q is per definition:• positive definite if Q(x) > 0 for all x 6= 0.• negative definite if Q(x) < 0 for all x 6= 0.• positive semidefinite if Q(x) _ 0 for all x 6= 0.• negative semidefinite if Q(x) _ 0 for all x 6= 0.• indefinite if Q(x) assumes both positive and negative values.The classification of a quadratic form is often carried over to the matrix of the form. Thus a positivedefinite matrix A is a symmetric matrix for which the quadratic form xTAx is positive definite.

1.3 Geometric View of Principal Axes

When Q(x) = Ax, where A is an invertible 2×2 symmetric matrix, and c is a constant,

then the set of

all x such that Ax = c corresponds to an ellipse (or circle) or a hyperbola. An ellipse is

described bythe following equation in standard form: x21a2 + x22b2 = 1 (a > b > 0), where a is the semi-mayor axes and bis the semi-minor axes. A hyperbola is described by the following equation in standard form: x21a2 − x22b2 = 1(a > b > 0), where the asymptotes are given by the equations x2 = ±bax1.Theorems:

1. If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.2. An n × n matrix A is orthogonally diagonalizable if, and only if A is a symmetric matrix.13. The Spectral Theorem for Symmetric Matrices: An n × n symmetric matrix A has thefollowing properties:(a) A has n real eigenvalues, counting multiplicities.(b) The dimension of the eigenspace for each eigenvalue _ equals the multiplicity of _ as a root ofthe characteristic equation.(c) The eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding todifferent eigenvalues are orthogonal.(d) A is orthogonally diagonalizable.4. The Principal Axes Theorem: Let A be an n×n symmetric matrix. Then there is an orthogonalchange of variable x = Py, that transforms the quadratic form xTAx into a quadratic form yTDywith no cross-product term.5. Quadratic Forms and Eigenvalues: Let A be an n × n symmetric matrix. Then a quadraticform xTAx is:(a) positive definite if, and only if the eigenvalues of A are all positive.(b) negative definite if, and only if the eigenvalues of A are all negative.(c) positive semidefinite if, and only if one eigenvalue of A is 0, and the others are positive.(d) negative semidefinite if, and only if one eigenvalue of A is 0, and the others are negative.(e) indefinite if, and only if A has both positive and negative eigenvalues

Question:

Show that the form 4 -8 +5 is positive definite where as is

not positive definite.

Solution:

A =4 -8 +5

=4( -2 - + )

=4( -2 + )+

=2( - + >0 x≠0This shows that it is a positive definite. B =

=

=

=3 +10 +2

= 2( +5 +3

=2( + +3 -

=2( + - >0

But, = -18 <0

This shosws that it is not positive definite.

References: www.aerostudents.com/files/.../symmetricMatrices QuadraticForm s.pdf https://kb.osu.edu/dspace/bitstream/1811/22234/.../V074N5_273.pdf Higher engi. Mths By B.S.Grewal

Advanced engi. Mths By R K Jain & S R K Lyengar