3. Types of Matrices:
1.
Identity
2.
Inverse Matrices
3.
Diagonal
4.
Symmetric
5.
Orthagonal
Identity
Matrix
An identity matrix is
a square matrix of dimensions (n, n) having '1' across its main diagonal and
'0' everywhere else. It is usually represented as 'In'
Example: The matrix shown below illustrates a (3, 3) Identity matrix
Python program
import numpy as np
#Creating
an Identity matrix of size 2 using np.identity() function
identity_matrix_1
= np.identity(2)
print("Identity_matrix 1\n",identity_matrix_1)
Linear
algebra offers a powerful tool called matrix inversion
To
describe matrix inversion, we first need to define the concept of an identity
matrix. An identity matrix is a matrix that does not change any vector when we
multiply that vector by that matrix. We denote the identity matrix that
preserves n-dimensional vectors as In.
The
structure of the identity matrix is simple: all of the entries along the main
diagonal are 1, while all of the other entries are zero.
The
matrix inverse of A is denoted as A−1 , and it is defined as the matrix such
that A−1A = In.
Ax=b=A-1x
Some
special kinds of matrices and vectors are particularly useful.
Diagonal Matrix
A Square Matrix (D) is called a Diagonal Matrix, if
D has zeros outside the main diagonal or principal diagonal. The Main
diagonal or the principal diagonal are the elements on the diagonal
that runs from the top left to bottom right.
Example: The matrix 'A' shown below illustrates a (3, 3) Diagonal matrix
We
have already seen one example of a diagonal matrix: the identity matrix,
where all of the diagonal entries are 1. We write diag(v) to denote a square
diagonal matrix whose diagonal entries are given by the entries of the vector
v. Diagonal matrices are of interest in part because multiplying by a diagonal
matrix is very computationally efficient. To compute diag(v)x, we only need to
scale each element xi by vi. In other words, diag(v)x = v . x. Inverting a
square diagonal matrix is also efficient. The inverse exists only if every
diagonal entry is nonzero, and in that case, diag(v)−1 = diag([1/v1, . . . ,
1/vn ]T)
Example Program in Python:
import numpy as np
# Creating a diagonal matrix with diagonal elements as
(1,2,3)
diagonal_matrix = np.diag((1,2,3))
print(diagonal_matrix)
# Creating a diagonal matrix with a range of values
matrix_range= np.diag(np.arange(1,6,2))
print(matrix_range)
Symmetric
A
symmetric matrix is any matrix that is equal to its own transpose: A = AT
Python Program
import numpy as np
#Creating
matrix A
A = np.array([[2,3,1], [3,4,-1], [1,-1,1]])
print("A:\t"
, A)
#
Finding the Transpose of the matrix
transposed_matrix
= A.transpose()
print("Transpose
of A:\n" , transposed_matrix)
comparison = (A == transposed_matrix)
#Checking
if all the elements in the matrix comparision is true
equal_arrays
= comparison.all()
print(equal_arrays)
Triangular
Matrix
A triangular
matrix can be either a lower triangular or an upper triangular matrix.
A lower
triangular matrix is a square matrix in which all the elements above
the main diagonal are zero.
Example: L
is a lower triangular matrix of dimension (3, 3)
More on Vectors :
import numpy as np
vector_1 = np.array([1,2,3])
vector_2
= np.array([1,0,3])
print("Vector
1:\n",vector_1)
print("Vector
2:\n",vector_2)
#
Finding product of vector of same dimensions using inner() function
inner_product_1
= np.inner(vector_1,vector_2)
print("Inner
Product of Vector 1, Vector 2:\n",inner_product_1)
Orthogonal vectors
If
the inner product of two non-zero vectors v1 and v2 is
zero, that is
then, the vectors v1 and v2 are called orthogonal vectors.
Example: Let
v1 and v2 be two vectors as follows:
import numpy as np
#Creating
vectors
Vector_1
= np.array([[3],[-1],[2]])
Vector_2
= np.array([[2],[4],[-1]])
print("Vector
1\n",Vector_1)
print("Vector
2\n",Vector_2)
#Finding
the transpose of Vector_1
trans
= np.transpose(Vector_1)
#Finding
the dot product
result
= np.dot(trans,Vector_2)
print("Dot
Product\n",result)
4.
Norm
L
p
||x||p
= (∑|xi|p )1/p
Norms,
including the L p norm, are functions mapping vectors to
non-negative values. On an intuitive level, the norm of a vector x measures the
distance from the origin to the point x.
More
rigorously, a norm is any function f that satisfies the following properties:
•
f (x) = 0 ⇒
x = 0
•
f (x + y) ≤ f(x) + f (y) (the triangle inequality)
•
∀α ∈ R, f (αx) = |α|f(x)
The
L 2 norm, with p = 2, is known as the Euclidean norm. It is simply
the Euclidean distance from the origin to the point identified by x. The L 2
norm is used so frequently in machine learning that it is often denoted simply
as ||x||, with the subscript 2 omitted. It is also common to measure the size
of a vector using the squared L 2 norm, which can be calculated simply as xTx.
The
L1 norm may be simplified to ||x||1 = (∑|xi| )
________
PPTs
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1. Introduction & Linear Algebra.
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