$1\mathrm{.}1$ Load and prepare the data set $[10$ pts$].$

There are a total of $1288$ images in this data set and each image is a $5037$ matrix.

Images are typically converted to a single vector of pixel values and are stacked as

columns of the matrix. Therefore, this image data set can be stored as a $18501288$

matrix, where the $1850$ is due to $5037.$ The dataset images.mat is available on

Canvas. Load this file to your Matlab workspace:

$>>$ B $=$ l o a d $($ images $.$ mat $).$ image data $$ ;

and make sure the matrix B is $18501288.$

The original images are in color; we will convert it to grayscale for convenience. This

scales the matrix such that all elements are in $[0,1].$ You can do this using the

mat$2$gray$()$ function in Matlab as:

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$>>$ B $=$ mat$2$gray $($ B $)$ ;

Now you should have the matrix ready for further analysis.

Deliverables $1\mathrm{.}1.$

$$ Plot the first $10$ images of the data set to confirm it matches Figure $1.$

To plot images, you must $(1)$ reshape each column of B into a $5037$ matrix us$-$

ing the reshape$()$ function: $>>$ firstimage $=$ reshape$($B$($:$,1),[37,50]),(2)$

then, use imshow$()$ function to plot each image using the reshaped column: $>>$

imshow$($firstimage$),(3)$ to plot an array of images as in Figure $1,$ use the subplot$()$

function in Matlab.

$1\mathrm{.}2$ Compute and plot the eigenvalues $[15$ pts$].$

We will first compute symmetric matrix from the data set; for this compute A $=$ BB$,$

which should be a $18501850$ square matrix. Then, use the eig$()$ function in Matlab

to compute both eigenvalues and eigenvectors.

Deliverables $1\mathrm{.}2.$

$$ The eig$()$ function will produce $1850$ eigenvalues, but you would see that

$18501288=562$ eigenvalues are nearly zero. These $562$ eigenvalues and

the corresponding eigenvectors are due to numerical error and should be re$-$

moved. Try the index slicing in MATLAB to truncate the arrays of eigenvalues

and eigenvectors that you obtain from eig$().$

$$ Plot the remaining $1288$ eigenvalues on a log$-$log plot in decreasing order of mag$-$

nitude $($that is$,$ your plot should show a curve that is decreasing in magnitude

from left to right$).$

$$ Explain what the large and small eigenvalues mean.

Caution: Matlab computes eigenvalues and eigenvectors in reverse order. That is$,$ the

first column of U corresponds to the smallest eigenvalue $($which is the first diagonal

element at $(1,1)$ in D$)$ and so on$.$ Therefore, you must reverse this order before

plotting. Check out flip$()$ in Matlab.

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$1\mathrm{.}3$ Plot the eigenvectors $[10$ pts$].$

Deliverables $1\mathrm{.}3.$

$$ From the eigen decomposition you did in the previous part, plot the first $10$

eigenvectors corresponding to the $10$ largest eigenvalues. Plot as images. Briefly

explain what you observe from the plots.

$$ Plot the last $10$ eigenvectors corresponding to the smallest $10$ eigenvalues. Plot

as images. Explain the difference with the plot in the previous step

I really need help on $1\mathrm{.}3.$ I cannot seem to get the $2$ additional plots. Please help me find the code using MATLAB. Thank you