1 Week 3: Vectors and Matrices (Part III) READING: 2.2 – 2.4 EECS 1541 -- Introduction to...

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EECS 1541 -- Introduction to Computing for the Physical Sciences

Vectors and Matrices as Function Arguments

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• In MATLAB, an entire vector or matrix can be passed as an argument to a function such as sum, prod

>> v = 1:5;>> sum(v)

ans =

15

• For a vector, the function will be evaluated on every element.

• Example:

>> v = 1:5;>> prod(v)

ans =

120

Means 1 + 2 + 3 + 4 + 5

Means 1 * 2 * 3 * 4 * 5


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• For matrices, the sum and prod functions operate on every individual column.

• Hence, if a matrix has dimensions: r x c, the result for the sum and prod functions will be a 1 x c row vector

>> A = [1:3; 2:4; 5:7];>> sum(A)

ans =

8 11 14

• Example:

from 3 + 4 + 7

from 2 + 3 + 6

from 1 + 2 + 5


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>> A = [1:3; 2:4; 5:7];>>

• Example: How would you compute the sum of all the entries in A?


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• Some other commonly used functions in array include:

functions descriptions Examples in vectors:

min Returns the smallest number

>> v = [1 2 3 4];>> min(v)ans = 1

max Returns the largest number

>> v = [1 2 3 4];>> max(v)ans = 4

cumsum Returns the cumulative sum

>> v = [1 2 3 4];>> cumsum(v)ans = 1 3 6 10

cumprod Returns the cumulative product

>> v = [1 2 3 4];>> cumprod(v)ans = 1 2 6 24


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>> A = [1:3; 2:-1:0; 5:7];

• Example: How would you determine the largest number in the following matrix?


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• Example: >> A = [1:3; 2:4; 5:7];>> cumsum(A)

ans =

1 2 3 3 5 7 8 11 14

• Hence, the resulting matrix will have the same dimension as the input matrix.

• When the cumsum and cumprod functions are used in matrices, they return the cumulative sum or product of every column:

Matrix transpose

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• Example: >> A = [1 2; 3 4; 5 6]

A =

>> transpose_A = A’

transpose_A =

1 3 5 2 4 6

• If A is an m x n matrix, then the transpose of A is an n x m matrix, where the row vectors of A are written as column vectors

1 2 3 4 5 6

Scalar multiplication

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• Example of a vector: >> v = [1 3 5 7];

>> v*3

ans = 3 9 15 21

• Numerical or arithmetic operations can be performed on every element in the entire vectors or matrices

Scalar multiplication

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• Example of a matrix: >> A = [1 3; 5 7; 2 4];

>> A*3

ans = 3 9 15 21 6 12

Scalar addition and subtraction

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• Example: >> A = [1 3; 5 7; 2 4];

>> A + 2

ans = 3 5 7 9 4 6

Add every element by 2

Array operations: addition and subtraction

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• You can perform element-by-element arithmetic with two arrays of the same size

>> v1 = [1 2 3];>> v2 = [4 5 6];>> v1 + v2

ans = 5 7 9

• Example:

Adding two vectors


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>> A1 = [1 2 3; 2 2 5];>> A2 = [1 0 1; 3 6 1];

>> A1 + A2

ans = 2 2 4 5 8 6

• Example:

Adding two matrices


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>> A1 = [1 2 3; 2 2 5];>> A2 = [1 0 1; 3 6 1];

>> A1 - A2

ans = 0 2 2 -1 -4 4

• Example:

Subtracting two matrices


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>> A1 = [1 2 3; 2 2 5];>> A2 = [1 0 1; 3 6 1];>> A2(2,:)=[]

>> A1 - A2

ans = Error using - Matrix dimensions must agree.

• Example:

Subtracting two matrices with different dimensions

Matrix Multiplication

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• Multiplication between two matrices works as follows:

• To multiply a matrix A by a matrix B to result in a matrix C, the number of columns of A must equal to the number of row in B

mxpnxpmxn CBA

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6

212

24

1

4*

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21• Example:

2 x 2 2 x 1 2 x 1

the inner dimensions must be the same

Matrix Multiplication

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• Example:

>> A = [1 2; 3 2];>> B = [4 1]’;

>> A*B

ans = 6 14

Multiplying two matrices

Vector Multiplication

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1111 xnxcxnr svv

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1

0

2

*421

• Example:

1 x 3 3 x 1 1 x 1

nxnxnrnxc svv 11


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• Example:

>> A = [1 2 4];

>> B = [2; 0; 1];

>> A*B

ans = 6

Multiplying two vectors

1 x 3 vector

3 x 1 vector


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• Example:

>> A = [1 2 4];

>> B = [2; 0; 1];

>> B*A

ans = 2 4 8 0 0 0 1 2 4

1 x 3 vector

3 x 1 vector

Multiplying two vectors

3 x 3 matrix

Array operations: Multiplication

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• How do we perform element-by-element multiplication or division between 2 vectors or 2 matrices?

• Example: >> v1 = [1:6]

v1 = 1 2 3 4 5 6

>> v1*v1

MATLAB will interpret this one as multiplying a 1 x 6 vector with a 1 x 6 vector

>> Error using * Inner matrix dimensions must agree.


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• To perform element-by-element multiplication-based operation between multiple vectors, a “dot” must be placed in front of the operator

operators descriptions Examples in vectors:

.* element-by-element multiplication

>> v = [1 2 3 4];>> v.*vans = 1 4 9 16

./ element-by-element division

>> v = [1 2 3 4];>> v./vans = 1 1 1 1

.^ element-by-element exponentiation

>> v = [1 2 3 4];>> v.^3ans = 1 8 27 64


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• Recall from LAB 2, when we plot a function that involves multiplication among sub-functions, we are performing element-by-element multiplication operation between multiple vectors

• Example, to plot the function: y = xex

>> x = [0:0.1:0.5]

x = 0.0 0.1 0.2 0.3 0.4 0.5

>> exp(x)

ans = 1.000 1.1052 1.2214 1.3499 1.4918 1.6487

>> y = x.*exp(x)

y = 0.000 0.1105 0.24438 0.4050 0.5967 0.8244

A row vector for the independent variable


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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


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• To perform element-by-element multiplication-based operation between 2 matrices, the dimensions must be the same

>> A = [1 2; 3 2];>> B = [2 0; 1 4];

>> A.*B

ans = 2 0 3 8

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21A

41

02B

Multiply each element in the same position of A and B

4*21*3

0*22*1Final result is:


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• NOTE: the difference between element-by-element multiplication and the actual matrix multiplication in this example since we have two 2 x 2 matrices

>> A = [1 2; 3 2];>> B = [2 0; 1 4];

>> A.*B

ans = 2 0 3 8

>> A = [1 2; 3 2];>> B = [2 0; 1 4];

>> A*B

ans = 4 8 8 8

Matrix multiplicationElement-by-element multiplication


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>> A = [1 2; 3 2];>> B = [2 0; 1 4];

>> B.*A

ans = 2 0 3 8

>> A = [1 2; 3 2];>> B = [2 0; 1 4];

>> B*A

ans = 2 4 13 10

Matrix multiplicationElement-by-element multiplication

A.*B = B.*A A*B ≠ B*A


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• Example: >> v = [2:2:8];>> v.^2/2

ans = 2 8 18 32

• Example: >> A = [ones(1,3); 1:3];>> A.^2’

ans =

1 1 1 41 9

^ has higher precedence than /

1 Week 3: Vectors and Matrices (Part III) READING: 2.2 – 2.4 EECS 1541 -- Introduction to...

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