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Reduce the following quadratic forms to canonical form by means of a modal matrix; are they...
Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:
Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an orthogonal matrix. i) Write Q as X T AX . ii) Find the Eigenvalues of A. Is the quadratic form...
Reduce to canonical form the following quadratic forms on R 3 : a) Q(x) = x12...
Reduce to canonical form the following quadratic forms on R 3 : a) Q(x) = x12 + x 22 + 3x32+ 4x1x2 + 2x1x3 + 2x2x3; b) Q(x) = 2x1x2 − 6x2x3 + 2x1x3. The form Q = a1y12+ a2y22 + · · · + anyn2 , where y1, y2, . . . , yn are new unknowns, are called canonical
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e....
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
[1 2 2 3 2 3 (a) Factorize the matrix A =| 2 | into elementary...
[1 2 2 3 2 3 (a) Factorize the matrix A =| 2 | into elementary matrices. (b) Write the condition for positive and negative definite quadratic forms. Reduce the quadratic form q=4x7+3x2-x?+2x2x3-4x3x1+4x1x2 to the canonical form. Hence find rank, index and signature of q. Write down the corresponding equations of transformation.
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric...
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form,...
1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form, convert it to a canonical form and determine the type of a curve to which it corresponds. 2. [16 marks] Find the spectral matrix and the corresponding modal matrix for -5 0 157 B = 3 4 -9. Write down the formula that needs to be used to diago- -5 0 15 nalise matrix B, but do not perform matrix multiplications.
i dont understand this problem. please show how to solve all parts using MATLAB. thank you. State-Space Representation and Analysis csys canon(sys,type) compute a canonical state-space realization...
i dont understand this problem. please show how to solve all
parts using MATLAB. thank you.
State-Space Representation and Analysis csys canon(sys,type) compute a canonical state-space realization type 'companion': controllable canonical form type modal: modal canonical form poles of a system controllability matrix observability matrix eig(A) ctrb(A,B) obsv(A,C) -7 L-12 0 EX A 2C-ioD0 uestions () Define the system in the state-space form (2) Determine the stability of the system (3) Determine the controllability and the observability of the system....
using the technique pictured, find the controllable canonical form of In this section we shall first...
using the technique pictured, find the controllable
canonical form of
In this section we shall first review technlqes into canonical forms. Then we shall review the invariance property of the Consider conditions for the controllability matrix and observability matrix orming State-Space Equations Into Canonical forms. crete-time state equation and output equation x(k +1) Gx(k) + Hu(k) y(k) Cx(k) + Du(k) We shall review techniques for transforming the state-s (6-30) (6-31) pace equations defined by Equations (6-30) and (6-31) into the...
1. (10 points) Consider quadratic form q ) = ? Aš where: 1 0 C A=...
1. (10 points) Consider quadratic form q ) = ? Aš where: 1 0 C A= -2 3 -2 T=Y -3 -4 -5 ܠܛ 2 (a) Find a symmetric matrix Q such that q(7) = 2 Q7. (b) Determine whether the quadratic form q is positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite.
Classify the quadratic form below. Then make a change of variable, x= Py, that transforms the...
Classify the quadratic form below. Then make a change of variable, x= Py, that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. -5x2 - 12x4x2 What is the most precise classification for the quadratic form? O A. Positive definite O B. Indefinite O C. Negative semidefinite OD. Positive semidefinite O E. Negative definite The new quadratic form is y'Dy=N.
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
[1 2 2 3 2 3 (a) Factorize the matrix A =| 2 | into elementary matrices. (b) Write the condition for positive and negative definite quadratic forms. Reduce the quadratic form q=4x7+3x2-x?+2x2x3-4x3x1+4x1x2 to the canonical form. Hence find rank, index and signature of q. Write down the corresponding equations of transformation.
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form, convert it to a canonical form and determine the type of a curve to which it corresponds. 2. [16 marks] Find the spectral matrix and the corresponding modal matrix for -5 0 157 B = 3 4 -9. Write down the formula that needs to be used to diago- -5 0 15 nalise matrix B, but do not perform matrix multiplications.
i dont understand this problem. please show how to solve all
parts using MATLAB. thank you.
State-Space Representation and Analysis csys canon(sys,type) compute a canonical state-space realization type 'companion': controllable canonical form type modal: modal canonical form poles of a system controllability matrix observability matrix eig(A) ctrb(A,B) obsv(A,C) -7 L-12 0 EX A 2C-ioD0 uestions () Define the system in the state-space form (2) Determine the stability of the system (3) Determine the controllability and the observability of the system....
using the technique pictured, find the controllable
canonical form of
In this section we shall first review technlqes into canonical forms. Then we shall review the invariance property of the Consider conditions for the controllability matrix and observability matrix orming State-Space Equations Into Canonical forms. crete-time state equation and output equation x(k +1) Gx(k) + Hu(k) y(k) Cx(k) + Du(k) We shall review techniques for transforming the state-s (6-30) (6-31) pace equations defined by Equations (6-30) and (6-31) into the...
1. (10 points) Consider quadratic form q ) = ? Aš where: 1 0 C A= -2 3 -2 T=Y -3 -4 -5 ܠܛ 2 (a) Find a symmetric matrix Q such that q(7) = 2 Q7. (b) Determine whether the quadratic form q is positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite.
Classify the quadratic form below. Then make a change of variable, x= Py, that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. -5x2 - 12x4x2 What is the most precise classification for the quadratic form? O A. Positive definite O B. Indefinite O C. Negative semidefinite OD. Positive semidefinite O E. Negative definite The new quadratic form is y'Dy=N.
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