Math 3607: Homework 5 2022
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Math 3607: Homework 5
2022
TOTAL: 30 points
• Problems marked with r are to be done by hand; those marked with 旦 are to be solved using a computer.
• Important note. Do not use Symbolic Math Toolbox. Any work done using sym or syms will receive NO credit.
• Another important note. When asked write a MATLAB function, write one at the end of your live script.
1. (Using eig; FNC 7.2.3) 旦 Use eig to find the EVD of each matrix. Then for each eigenvalue λ, use the rank command to verify that λI ´ A is singular.
A “ –(»—)´
´(´)
´
fl(fiffi) , B “ –(»—)´(´)
´(´)
´(´)
fl(fiffi) , C “ –(»———)
fl(fiffiffiffi) .
2. (Spectra and pseudospectra; Adapted from FNC 7.2.7.) 旦 The eigenvalues of Toeplitz matrices, which have a constant value on each diagonal, have beautiful connections to complex analysis. Define six 64 ˆ 64 Toeplitz matrices using
|
z = zeros (1,60); A{1} = toeplitz ( [0,0,0,0, z], [0,1,1,0,z] ); A{2} = toeplitz ( [0,1,0,0, z], [0,2i,0,0, z] ); A{3} = toeplitz ( [0,2i,0,0, z], [0,0,1,0 .7, z] ); A{4} = toeplitz ( [0,0,1,0, z], [0,1,0,0,z] ); A{5} = toeplitz ( [0,1,2,3, z], [0,- 1,-2,0, z] ); A{6} = toeplitz ( [0,0,-4,-2i , z], [0,2i ,- 1,2, z] ); |
(The variable A constructed hereinabove is a cell array and it contains all six matrices defined above. To access any one of them, simply use A{#}.) For each of the six matrices, do the following.
(a) Plot the eigenvalues of A{#} as red dots in the complex plane. (Set ’MarkerSize’ to be 3.)
(b) Let E and F be 64 ˆ 64 random matrices generated by randn. On top of the plot from part (a), plot the eigenvalues of the matrix A ` εE ` iεF as blue dots, where ε “ 10 ´3 . (Set ’MarkerSize’ to be 1.)
(c) Repeat part (b) 49 more times (generating a single plot).
Arrange all six plots in a 3 ˆ 2 grid using subplot. Make sure all figures are drawn in 1:1 aspect ratio.
3. (Recursively defined sequences) r The Pell numbers 0, 1, 2, 5, 12, 29, 70, 169, 208, 985, . . . are defined recursively by
Pn “ &1 if n “ 1
’’%2Pn ´ 1 ` Pn ´2 otherwise
Using an EVD of a suitable 2-by-2 matrix, find the general formula for the kth Pell number.
4. (Singular values by hand) r 旦 Calculate the singular values of
A “ 
ffi
– ´1 ´1fl
by solving a 2 ˆ 2 eigenvalue problem. Confirm your answer using MATLAB.
5. (Vandermonde matrix, SVD, and rank) 旦 Let x be a vector of 1000 equally spaced points between 0 and 1, and let An be the 1000 ˆ n Vandermonde-type matrix whose pi, jq entry is xi(j) ´ 1 for j “ 1, . . . , n.
(a) Print out the singular values of A1 , A2, and A3 .
(b) Make a semi-log plot of the singular values of A25 .
(c) Use rank to find the rank of A25 . How does this relate to the graph from part (b)? You may want to use the help document for the rank command to understand what it does.
2022-07-19