Please help with A.Thank you. 1. Let G have Laplacian matrix2-1-12-1-1-13-1-1L =|-13-1-1-1-13-1-1-12-1 -1 -13A. Use a matrix calculator to find the eigenvalues of L; there should besome pairs of them that have the same value. List them in orderIt's fine (suggested, actually) that you use decimal approximationsrather than exact values.B. Use a matrix calculator to find eigenvectors v and w corresponding to1z and 13. It's fine if you use decimal approximations for these.Compute the vectorv.Wz = w-V.V. VThis vector z is orthogonal to v.C. Let x = v and y = z and plot the pointsVzzVv.v(x1, Yı), (x2, Y2), (X3, Y3), (X4, Y4), (X5, Ys), (x6, Y6), (X7,Y7)and for each edge ij of G, draw the segment joining (xị, yı) to(Xj, Y).The result in C should be a "nice" drawing of G, in the sense thatadjacent vertices are close together.D. Do the same process in parts B and C for the eigenvectorscorresponding to l6 and 17, the two largest eigenvalues.E. The end result in part D should cause adjacent vertices to be drawn farapart and give you an idea of how to assign colors to the vertices todelermine the chromalic number of G. Whal is this chromalicnumber?

(A) By the matrix calculator the eigenvalues of L are 0<1.381966=1.381966<2.5857864<3.618034=3.618034<5.4142136 Comparing this by 0≤λ2≤λ3≤λ4≤λ5≤λ6≤λ7, we get λ2=λ3=1.381966, λ4=2.5857864, λ5=λ6=3.618034, λ7=5.4142136. (B) Using matrix calculator the eigenvectors v and w corresponding to λ2 and λ3, i.e. corresponding to 1.381966 are as follows: v=-100.61803401-0.618034000 and w=1.6180340-1.6180340-10010.To compute the vector z=w-v.wv.vv. First we will find the value of v.w and v.v Now v.w=-100.61803401-0.618034000T 1.6180340-1.6180340-10010=-1.6180340+0-0.6180340+0+0+0+0=-2.2360680 and v.v=-100.61803401-0.618034000T -100.61803401-0.618034000=1+0+(0.6180340)2+1+(-0.6180340)2+0+0=2.7639320 Hence, z=w-v.wv.vv= 1.6180340-1.6180340-10010--2.23606802.7639320-100.61803401-0.618034000=1.6180340+(0.8090170)(-1)-1.6180340+0-1+(0.8090170)(0.6180340)0+(0.8090170)(1)0+(0.8090170)(-0.6180340)1+00+0z=0.8090170-1.6180340-0.49975240.8090170-0.500247610 This vector z is orthogonal to v because z.v=0.8090170-1.6180340-0.49975240.8090170-0.500247610T -100.61803401-0.618034000=-0.8090170+0-0.3088640+0.8090170+0.3091700+0+0=0.0005300⇒z.v≈0 (C) Let x=1v.vv and y=1z.zz Since v.v=-100.61803401-0.618034000T -100.61803401-0.618034000=1+0+(0.6180340)2+1+(-0.6180340)2+0+0=2.7639320⇒v.v=2.7639320=1.6625077 and z.z=0.8090170-1.6180340-0.49975240.8090170-0.500247610T0.8090170-1.6180340-0.49975240.8090170-0.500247610=(0.8090170)2+(-1.6180340)2+(-0.4997524)2+(0.8090170)^2+(-0.5002476)2+1+0=5.4270512⇒z.z=5.4270512=2.3296032 So, x=11.6625077-100.61803401-0.618034000=-0.601500900.37174800.6015009-0.371748000 and y=12.32960320.8090170-1.6180340-0.49975240.8090170-0.500247610=0.3472767-0.6945534-0.21452250.3472767-0.21473510.42925760 Plotting of the all the points of the form (xi,yi), where 1≤i≤7. Below figure is the required graph G

Math

Advanced Math

1. Let G have Laplacian matrix 2 -1 -1 -1 -1 -1 3 -1 -1 L =-1 3 -1 -1 -1 -1 3 -1 -1 -1 2 -1 -1 -1 3 A. Use a matrix calculator to find the eigenvalues of L; there should be some pairs of them that have the same value. List them in order It's fine (suggested, actually) that you use decimal approximations rather than exact values. B. Use a matrix calculator to find eigenvectors v and w corresponding to Az and A3. It's fine if you use decimal approximations for these. Compute the vector v.w z = w- V.V This vector z is orthogonal to v. C. Let x =v and y = z and plot the points (x1, y1), (X2, y2), (x3, y3), (X4, Y4), (X5, Ys), (X6, Y6), (X7, Y7) and for each edge ij of G, draw the segment joining (x, yi) to (Xj, Yj).

1. Let G have Laplacian matrix 2 -1 -1 -1 -1 -1 3 -1 -1 L =-1 3 -1 -1 -1 -1 3 -1 -1 -1 2 -1 -1 -1 3 A. Use a matrix calculator to find the eigenvalues of L; there should be some pairs of them that have the same value. List them in order It's fine (suggested, actually) that you use decimal approximations rather than exact values. B. Use a matrix calculator to find eigenvectors v and w corresponding to Az and A3. It's fine if you use decimal approximations for these. Compute the vector v.w z = w- V.V This vector z is orthogonal to v. C. Let x =v and y = z and plot the points (x1, y1), (X2, y2), (x3, y3), (X4, Y4), (X5, Ys), (X6, Y6), (X7, Y7) and for each edge ij of G, draw the segment joining (x, yi) to (Xj, Yj).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.

Question

Please help with A.

Thank you.

1. Let G have Laplacian matrix
2
-1
-1
2
-1
-1
-1
3
-1
-1
L =|-1
3
-1
-1
-1
-1
3
-1
-1
-1
2
-1 -1 -1
3
A. Use a matrix calculator to find the eigenvalues of L; there should be
some pairs of them that have the same value. List them in order
It's fine (suggested, actually) that you use decimal approximations
rather than exact values.
B. Use a matrix calculator to find eigenvectors v and w corresponding to
1z and 13. It's fine if you use decimal approximations for these.
Compute the vector
v.W
z = w
-V.
V. V
This vector z is orthogonal to v.
C. Let x = v and y = z and plot the points
Vzz
Vv.v
(x1, Yı), (x2, Y2), (X3, Y3), (X4, Y4), (X5, Ys), (x6, Y6), (X7,Y7)
and for each edge ij of G, draw the segment joining (xị, yı) to
(Xj, Y).
The result in C should be a "nice" drawing of G, in the sense that
adjacent vertices are close together.
D. Do the same process in parts B and C for the eigenvectors
corresponding to l6 and 17, the two largest eigenvalues.
E. The end result in part D should cause adjacent vertices to be drawn far
apart and give you an idea of how to assign colors to the vertices to
delermine the chromalic number of G. Whal is this chromalic
number?

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