1. Storing two-dimensional numerical data
Matrices
Storing two-dimensional numerical data
- next couple lectures: learn about matrices, one of core
structures in MATLAB (name from “Matrix Laboratory”)
- we’ll initially uses matrices as means for storing 2-D
numerical data
- what contexts you’ve seen matrices? HS algebra?
solving linear equations? rotating coordinates?
courses or applications?

2. Analyzing table data
level
1998
1999
2000
2001
2002
2003
2004
2005
advanced 7 9 15 18 20 24 29 35
proficient 17 27 28
needs
improvement 23 22 30 31
failing 52 53 45 25
- table shows statewide results of MCAS Test in Math
for Grade 10 students, over years , indicating
percentage of students who achieved each level of
performance
- MCAS stands for Massachusetts Comprehensive
Assessment System and evaluates public education in
state. Grade 10 students need to pass this exam in
order to graduate HS
Statewide results for MCAS Test in Mathematics, Grade 10
Matrices

3. Medical imaging
Matrices are particularly good for storing data that is inherently two-dimensional
For example, the illustrated MRI slice is obtained from a two-dimensional grid of brightness measurements registered by an array of light sensitive elements
- matrices also good for storing data that is
inherently 2-D, one example is images, such as
this MRI image
- we’ll begin with basic background on matrices
in MATLAB, then consider a couple examples
involving processing images, then more basics
- later in semester, you’ll build a tool for
visualizing MRI data in different ways
Matrices

4. Matrices: The basics A matrix is a rectangular array of numbers
We create a matrix of specific values with an assignment statement:
>> flowers = [ ; ; ]
flowers = 1 3 2 7 6 4 5 8
- tired of “nums” and “numbers” – with all new snow,
time to think spring, flowers
- suppose we have explicit values want to store in
matrix - list inside brackets as we did with vector
- each row of values is separated by a semi-colon
(note no semi-colon after last row)
- contents of each row can be written with or without
commas between values - we’ll write them without
- MATLAB prints values row-by-row without brackets
- visualize as 2-D grid of numbers (1st, 2nd, 3rd rows)
flowers
Matrices

5. Dimensions Each row must contain the same number of values!
nums = [ ; 6 8]
size function returns the number of rows and columns in a matrix:
>> dims = size(flowers)
dims =
3 4
>> rows = size(flowers, 1)
rows = 3
>> cols = size(flowers, 2)
cols = 4 1 3 2 7 6 4 5 8
flowers
- when creating matrix of explicit values, each row must
have same number of values - if not, error: dimensions
not consistent
- used length function to determine number of elements
in vector, we use “size” with matrices
- in first case, row and columns returned by size are
stored in a vector: [rows cols] - how refer to each
dimension? dims(1) or dims(2)
- another way to call size function: add second input
that specifies which dimension you want
- numEntries = prod(dims)
prod(size(flowers))
rows*cols
How could you determine the total number of elements in a matrix?
Matrices

6. Déjà vu all over again
Many computations can be performed on an entire matrix all at once
flowers = 2 * flowers + 1 1 3 2 7 6 4 5 8
flowers 3 7 5 15 13 9
- just like vectors…
- arithmetic operations that combine matrix and
numbers - same operations applied to every
element of matrix, producing matrix of results
- final rows 2 and 3:
flowers
Matrices

7. Element-by-element matrix addition
sumFlowers = flowers + addOns 3 7 5 15 13 9 11 17 2 1 3 4 +
flowers
addOns 5 8 19 15 13 =
- also just like vectors, can perform arithmetic operations
between two matrices, element-by-element
- with old friend .* - use ./ for element-by-element
division and .^ to raise each element to power, e.g.
>> squares = flowers.^2
- what do you think * does? matrix multiplication
error here: (3 x 4) * (3 x 4) (inner dimensions error)
- important: to perform element-by-element operations
on two matrices, the matrices must be the same size,
i.e., same number rows and same number columns
sumFlowers
How do you perform element-by-element multiplication?
Matrices

8. Vectors are matrices… … with one row or column
rowNums 1 2 3
rowNums = [ ]
rowNumsSize = size(rowNums)
colNums = [1; 2; 3]
colNumsSize = size(colNums)
length function returns the largest dimension
rowNumsSize 1 3 1
colNums 2 3
- just learned that similar operations can be
performed on matrices and vectors – not surprizing –
vectors are just matrices with one row or col
- we distinguish with names vector/matrix, but to
MATLAB, all same
- create colNums with semi-colon after each number,
stored in column vector with dimensions 3 x 1
>> num = 2;
>> size(num)
ans =
- for vector, value returned by length would correspond
to the number of elements in the vector
- we need to be more cautious about some of the
functions that we may want to use with matrices
colNumsSize 3 1
Matrices

9. Now I know what you’re thinking…
You probably think that we can use functions like sum, prod, min, max and mean in the same way they were used with vectors:
numbers = [ ; ]
totalSum = sum(numbers)
totalProd = prod(numbers)
minVal = min(numbers)
maxVal = max(numbers)
meanVal = mean(numbers) 1 3 2 4
numbers 4 1 2 3
- these are valid statements in MATLAB, but
don’t have quite the effect you’re probably
thinking…
Matrices

10. Hmmm… that’s not what I expected…
numbers = [ ; ]
totalSum = sum(numbers)
totalProd = prod(numbers)
minVal = min(numbers)
maxVal = max(numbers)
meanVal = mean(numbers) 1 3 2 4
numbers 4 1 2 3
totalSum 5 4 4 7
totalProd 4 3 4 12
- performs operations within columns, giving vector of results
for each column
- compute sum of all the numbers in matrix: sum(sum(numbers))
- how can you modify this statement to compute the product,
min, max and mean values of all the contents of a matrix?
- awkward, but the way things are in MATLAB…
- follow-up sum, prod, min, max, mean:
sum(numbers, 1) sums across the rows of the matrix
- equivalent to sum(numbers)
sum(numbers, 2) sums across cols - [10; 10] column vector
- prod and mean behave the same way (example)
- min(numbers, [], 1) min(numbers, [], 2)
- extra empty vector in middle, same with max
- min(numbers, 2) returns matrix of same size as
numbers whose entries are the minimum of the current
entry and 2, similarly with max(numbers, 2)
minVal 1 1 2 3
meanVal
2.5
2.0
2.0
3.5
maxVal 4 3 2 4
Matrices

11. Processing and displaying images
An image is a two-dimensional grid of measurements of brightness
We will start with images with brightness ranging from black (0.0) to white (1.0) with shades of gray in between (0.0 < b < 1.0)
- e.g. brightness measured with a digital camera
- black & white, or “gray-level” images
- represent images as 2-D matrix of numbers,
0.0 to 1.0
1887 Crew Team Wellesley College
Matrices

12. Creating a tiny image >> imshow(tinyImage) tinyImage
; …
; … ]
0.0
0.0
0.0
0.0
0.0
0.0
>> imshow(tinyImage)
0.0
0.5
0.5
0.5
0.5
0.0
0.0
0.5
1.0
1.0
0.5
0.0
- images are typically very large, at least 128 elements on a side
- very tiny image (6x6) with 3 levels of brightness - white (1.0)
in center, surrounded by medium gray (0.5) and black (0.0)
- function imshow displays image in figure window, see how looks
in MATLAB - while there, pursue mystery… ( next slide)
(use setupImages.m to create images for display)
- display functions from MATLAB Image Processing Toolbox
- imshow(tinyImage) (enlarge figure window by dragging corner)
- imshow(image) (corrupted mystery image)
tool bar: save, print, zoom in, zoom out, pan
128 x 128 image, each block is one element
- imtool(image) allows you to see numbers, dragging doesn’t
enlarge image - use menu, zoom in, zoom out, pan
- individual locations: pixels (picture elements),
lower left: x,y coords (matrix indices), brightness
(1,1): upper left corner to (128,128) lower right
- show pixel region tool
tinyImage
0.0
0.5
1.0
1.0 05
0.0
(not to scale)
0.0
0.5
0.5
0.5
0.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Matrices

13. A mystery: Who stole The Beast?
This very corrupted image was received by anonymous courier late last night
Let’s figure out what’s in it using the Image Processing Toolbox
>> imtool(image)
- what are all those bright specks (1.0)? Let’s get rid of them by
changing brightness at those locations to brightness of background
>> image(image == 1.0) = 0.45; (similar to vector)
- something emerging, but dark, low contrast. Let’s beef up contrast
- helps to know range of brightness values in image:
>> minV = min(min(image)) 
>> maxV = max(max(image)) 
- now expand range of brightness by shifting darkest brightness
value down to 0.0 and stretch out values so they span range 0-1
>> image = image -minV; check min/max again
>> minV= min(min(image)  0
>> maxV = (max(max(image)) 
- but we want max to be 1, so multiply each value in image by
1/ or about 18
>> image = image * 18;
>> imshow(image)  Sohie! (with measles…)
Matrices

14. Creating matrices with constant values
To create a matrix of all ones:
nums1 = ones(2,3)
nums2 = ones(1,5)
To create a matrix of all zeros:
nums3 = zeros(3,1)
nums4 = zeros(4,3) 1 1 1
nums1 1 1 1
nums2 1 1 1 1 1
- sometimes useful to create a matrix of particular size,
initially filled with constant value (trust me!)
- can use to create vectors as well, but still need to give
two dimensions
- suppose want to create initial matrix 2x4, filled
with 10’s?
>> tens = 10*ones(2,4)
nums4
nums3
Matrices

15. Indexing with matrices
Each row and column in a matrix is specified by an index
nums = [ ; ]
We can use the indices to read or change the contents of a location
val = nums(2,3)
nums(1,4) = 9
nums(1,end) = 9 1 3 7 4 8 5 2 6
nums
- first row, first column have index 1, indices
increment by 1 for successive rows/cols
- val = 2, the 4 in row 1 column 4 changes to 9
- keyword end, when provided as a row or column
index for a matrix, refers to the last index of the
row or column
Similar to vectors
Matrices

16. Time-out exercise Starting with a fresh copy of nums
what would the contents of nums and val be after executing the following statements?
nums(2,3) = nums(1,2) + nums(2,4)
nums(1,3) = nums(1,4) + nums(2,1)
val = nums(4,3) 1 3 7 4 8 5 2 6
nums
- 2 changes to 9
- 7 changes to 12
- last one is a trick, we are attempting to read the
contents of a location whose index is outside range
of indices of the matrix and an error is generated
Matrices

17. Auto expansion of matrices
>> nums = [ ; ]
>> nums(4, 7) = 3 1 3 7 4 8 5 2 6
nums
nums 1 2 3 4 5 6 7 1 3 7 4
- ability to expand a vector automatically can be
advantageous at times, but can also be a source of bugs
in your code. Beware!
- important note: when MATLAB automatically expands
matrix, allocates new memory space to store the larger
matrix and copies contents - this process takes time! This
is one reason why we create initial matrices of appropriate
size with the ones and zeros functions 1 2 8 5 2 6 3 4 3
Matrices

18. Analyzing table data
level
1998
1999
2000
2001
2002
2003
2004
2005
advanced 7 9 15 18 20 24 29 35
proficient 17 27 28
needs
improvement 23 22 30 31
failing 52 53 45 25
- table shows statewide results of MCAS Test in Math
for Grade 10 students, over years , indicating
percentage of students who achieved each level of
performance
- MCAS stands for Massachusetts Comprehensive
Assessment System and evaluates public education in
state. Grade 10 students need to pass this exam in
order to graduate HS
Statewide results for MCAS Test in Mathematics, Grade 10
Matrices

19. Indexing with colon notation
To refer to an entire column of a matrix, provide : as the first index and the column number as the second index
>> nums(:, 3)
ans = 3 8 13 18
To refer to an entire row of a matrix, provide : as the second index and the row number as the first index
>> nums(2, :)
nums 1 2 3 4 5 1 2 3 4 5 1 2 6 7 8 9 10 3 11 12 13 14 15 4 16 17 18 19 20
- suppose we have a table of information and
want to select entire column or row of our data
for analysis or for plotting (in a moment…)
- suppose nums(:, end) or nums(end, :)
Matrices

20. Plotting trends in performance levels
We begin our analysis by plotting the data for each performance level over the 8 years
% create matrices that store data and years
results = [ ; ...
; ...
; ... ];
years = [ ];
Each row of the table corresponds to a performance level. How do we plot the resulting trend over the given years?
>> plot(years, results(1, :), ‘b’); (advanced)
- since we’re drawing several plots on same graph,
it would be good to “hold on” until we’re done
- I like to draw thick lines, using LineWidth property
plot(years, results(1,:), ‘b’, ‘LineWidth’, 2)
Matrices

21. Plotting the data % plot the data for each performance level vs. years
hold on
plot(years, results(1,:), 'b‘, ‘LineWidth’, 2);
plot(years, results(2,:), 'g‘, ‘LineWidth’, 2);
plot(years, results(3,:), 'c‘, ‘LineWidth’, 2);
plot(years, results(4,:), 'r‘, ‘LineWidth’, 2);
hold off
xlabel('year‘)
ylabel('percentage of students‘)
title('MCAS results‘)
legend('advanced‘, 'proficient‘, 'improve‘, 'failing‘ );
- plots with different colors, all thick, and usual
labels, title, legend (strings in order plotted)
Matrices

22. Finally, ...
Suppose we want to print the change in results between 1998 and 2005 for each performance level…
How do we do this?
- can begin with creating column vector with
changes for all four levels:
totalChange = results(:, end) - results(:, 1)
Matrices

23. Printing changes in results
% print total change in results between 1998 and 2005
totalChange = results(:, end) - results(:, 1);
disp('Change in performance between 1998 and 2005:‘);
disp(['advanced: ' num2str(totalChange(1)) '%‘]);
disp(['proficient: ' num2str(totalChange(2)) '%‘]);
disp(['needs improvement: ' num2str(totalChange(3)) '%‘]);
disp(['failing: ' num2str(totalChange(4)) '%‘]);
- recall general form of the disp function is
disp([string1 string2 ... stringLast]);
- concatenates all the strings (note spaces)
- num2str converts number to string (if
omitted, MATLAB prints a box for the number)
Change in performance between 1998 and 2005:
advanced: 28%
proficient: 10%
needs improvement: 0%
failing: -37%
Matrices

24. Time-out exercise
For each year, compute a weighted sum of the four percentages, using a weight of 1 for “advanced”, 2 for “proficient”, 3 for “needs improvement” and 4 for “failing”*
overallPerformance =
Add a new row to the results matrix that stores these weighted sums
overallPerformance =
1 * results(1,:) + 2 * results(2,:) …
+ 3 * results(2,:) + 4 * results(4,:)
- hmmm, that looks vaguely familiar:
[ ] * [ … …
… ]
- our old friend matrix multiplication does the trick:
overallPerformance = (1:4) * results
- let’s take advantage of auto-expansion of matrix:
results(5,:) = overallPerformance
* The resulting sum can range from 100 (great!) to 400 (not so good…)
Matrices

25. More indexing with colon notation
We can use colon notation to refer to a range of indices within a column or row of a matrix
>> nums(1:3, 4)
ans = 4 9 14
>> nums(3, 3:5)
>> nums(2:3, 2:4)
nums 1 2 3 4 5 1 2 3 4 5 1 2 6 7 8 9 10 3 11 12 13 14 15
- suppose want to refer to subset of row or column,
or rectangular region of values within matrix
- nums(2:end, 3:end)
- suppose use in assignment:
nums(2:end, 3:end) = 10 4 16 17 18 19 20
Matrices

26. Conditional operations on matrices
A conditional expression can be applied to an entire matrix all at once producing a new matrix of the same size that contains logical values
ages = [ ; ; ];
teens = (ages >= 13) & (ages <= 19);
ages
teens 13 52 19 21 1 1
- recall that we used this idea when interpreting
the noisy image of Sohie:
image(image == 1.0) = 0.45 18 47 23 15 1 1 60 38 16 12 1
Matrices

27. Using logical vectors >> ages(teens) = 0 ages = 0 52 0 21
>> overTheHill = ages(ages>40)
overTheHill = 60 52 47 13 52 19 21
ages 18 47 23 15 60 38 16 12 1 1
teens
- in the overTheHill example, answers are selected
column-by-column, and all values are then collected
into a single column vector 1 1 1
Matrices

28. Time-out exercise
Given the original ages matrix, write two statements that each assign the variable numAdults to the total number of age values that are 18 or over
One statement should use sum and the other should use length
numAdults = sum(sum(ages >= 18))
numAdults = length(ages(ages>=18))
Matrices