Matrices and Vectors

Matrices
- It is a multi-dimensional array
- Dimension of matrix = number of rows x number of columns
Matrix.png
Elements Of Matrices
- Elements of the matrix can be accessed using the i or j the term
- i th term = row && j th term = column
Vectors
- It a a matrix with on column and ‘n’ rows.
In general, all our vectors and matrices will be 1-indexed. Note that for some programming languages, the arrays are 0-indexed.
Matrices are usually denoted by uppercase names while vectors are lowercase.
“Scalar” means that an object is a single value, not a vector or matrix.

Octave/Matlab Snippet

Code for matrices and vectors

% The ; denotes we are going back to a new row.
A = [1, 2, 3; 4, 5, 6; 7, 8, 9; 10, 11, 12]

% Initialize a vector
v = [1;2;3]

% Get the dimension of the matrix A where m = rows and n = columns
[m,n] = size(A)

% You could also store it this way
dim_A = size(A)

% Get the dimension of the vector v
dim_v = size(v)

% Now let's index into the 2nd row 3rd column of matrix A
A_23 = A(2,3)

Addition and Scalar Multiplication

Matrix Addition Subtraction
- Only matrix with same dimensions can be operated.
- Size of the output matrix would be same as the input matrix
Matrix Addition.png
Scalar Multiplication Division
- Scalar gets multiplied or divided to each and every element of the matrix
Scalar Multiplication.png
Combination of Operands
- Many operations can be combined Combination of operands.png

Octave/Matlab Snippet

Octave/Matlab commands for matrix addition and scalar multiplication.

% Initialize matrix A and B
A = [1, 2, 4; 5, 3, 2]
B = [1, 3, 4; 1, 1, 1]

% Initialize constant s
s = 2

% See how element-wise addition works
add_AB = A + B

% See how element-wise subtraction works
sub_AB = A - B

% See how scalar multiplication works
mult_As = A * s

% Divide A by s
div_As = A / s

% What happens if we have a Matrix + scalar?
add_As = A + s

Matrix Vector Multiplication

Example 1
- The result is a vector. The number of columns of the matrix must equal the number of rows of the vector.
Example 1.png
Details
- An m x n matrix multiplied by an n x 1 vector results in an m x 1 vector.
Details.png
Example 2

Example 2.png
Hypothesis Trick
- Hypothesis can be applied to many values efficiently by using matrix-vector multiplication
- Matrix would be the data set or the value on which hypothesis is to be applied
- Vector would be the hypothesis
- Output would be the predictions from the hypothesis
Hypothesis trick.png

Octave/Matlab Snippet

matrix-vector multiplication

% Initialize matrix A
A = [1, 2, 3; 4, 5, 6;7, 8, 9]

% Initialize vector v
v = [1; 1; 1]

% Multiply A * v
Av = A * v

Matrix Matrix Multiplication

Example 1
- multiply two matrices by breaking it into several vector multiplications and concatenating the result.
Ex 1.png
Details
- An m x n matrix multiplied by an n x o matrix results in an m x o matrix.
- To multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second matrix.
Detail.png
Example 2

Ex 2.png
Hypothesis Trick
- Multiple hypothesis can be calculated using matrix matrix multiplication Hypo Trick.png

Octave/Matlab Snippets

matrix-matrix multiplication

% Initialize a 3 by 2 matrix
A = [1, 2; 3, 4;5, 6]

% Initialize a 2 by 1 matrix
B = [1; 2]

% We expect a resulting matrix of (3 by 2)*(2 by 1) = (3 by 1)
mult_AB = A*B

% Make sure you understand why we got that result

Matrix Multiplication Properties

Commutative
- Matrices are not commutative
  - A _ B ≠ B _ A
Commutative.png
Associative
- Matrices are associative
  - (A _ B) _ C = A _ (B _ C)
Associative.png
Identity Matrix
- The identity matrix, when multiplied by any matrix of the same dimensions, results in the original matrix.
- When multiplying the identity matrix after some matrix (A∗I), the square identity matrix’s dimension should match the other matrix’s columns.
- When multiplying the identity matrix before some other matrix (I∗A), the square identity matrix’s dimension should match the other matrix’s rows.
Identity Matrix.png

Octave/Matlab Snippet

% Initialize random matrices A and B
A = [1,2;4,5]
B = [1,1;0,2]

% Initialize a 2 by 2 identity matrix
I = eye(2)

% The above notation is the same as I = [1,0;0,1]

% What happens when we multiply I*A ?
IA = I*A

% How about A*I ?
AI = A*I

% Compute A*B
AB = A*B

% Is it equal to B*A?
BA = B*A

% Note that IA = AI but AB != BA

Inverse and Transpose

Inverse
- Multiplying by the inverse results in the identity matrix.
- A non square matrix does not have an inverse matrix.
- Matrices that don’t have an inverse are singular or degenerate.
Inverse.png
Transpose
- The transposition of a matrix is like rotating the matrix 90° in clockwise direction and then reversing it.
Transpose.png
Octave/Matlab Snippet
- compute inverses of matrices in octave with the pinv(A) function and in Matlab with the inv(A) function.
- We can compute transposition of matrices in matlab with the transpose(A) function
```
% Initialize matrix A
A = [1,2,0;0,5,6;7,0,9]

% Transpose A
A_trans = A'

% Take the inverse of A
A_inv = inv(A)

% What is A^(-1)*A?
A_invA = inv(A)*A
```