Linear Algebra Python Flashcards
Initialize a vector
- v = [3,2]
- v1 = np.array([3,2])
Addition of two vectors
v1 + v2
Scalar multiplication
2 * v
Transpose of a vector
- np.transpose(v)
- v.T
- v.T.T
Dot product approaches
- np.dot(v1, v2)
- np.matmul(v1, v2)
- sum(np.multiply(v1, v2))
Create one random number
np.random.rand/ np.random.randn
Create random vector of size 10
np.random.randn(10)
create a 3 * 1 matrix of random numbers
np.random.randn(3,1)
Get the dimensions of a matrix
mat.ndim
Get the number of elements in a matrix
mat.size
Get the size of the matrix - 4X6
mat.shape
Just get the number of rows of a matrix
mat.shape[0]
Just get the number of columns of a matrix
mat.shape[1]
Get the first column of a matrix
mat[: , 0]
Get the first and third columns of a matrix
mat[: , [0,2]]
Get the one to third columns of a matrix
mat[: , 0:3]
Repeat 1st column twice in a matrix
mat[:, [0,0]]
Get the first row of a matrix
mat[0, :]
Get the first and third rows of a matrix
mat[[0,2], :]
Get 1 to third rows in a matrix
mat[0:3 , :]
Repeat 1st row twice in a matrix
mat[[0,0], :]
Repeat 1st row n times in a matrix
mat[[0] * n , :]
Create a zero vector of size 4
np.zeros(4) # 4 row vector
convert 3X1 matrix to 3x1 vector
- mat = np.random.randn(3,1)
- arr = np.asarray(mat) # 2D array
- vector = np.squeeze(arr) # 1d array
magnitude of a vector
- np.linalg.norm(v)
- np.sqrt(sum(np.multiply(v,v)))
- np.sqrt(np.dot(v,v))
Angle between two vectors
theta = np.arccos((np.dot(a,b)) / (np.linalg.norm(a) * np.linalg.norm(b)))
How to know if two vectors are orthogonal
If the dot product of the two vectors is 0
How to create just positive random number
np.random.rand()
create duplicate dependent vector from v1
np.random.rand() * v
absolute value of a number
np.abs(-1)
Perform hadamard multiplication
- np.multiply(v1, v2)
- v1 * v2
Outer product
np.outer(v1, v2)
cross product of vectors
np.cross(v1, v2)
create complex number
np.complex(3,4) = 3 + 4j
Create complex vector
np.array([3, 4i, np.complex(5,2), 2-5i])
np.transpose(3 + 4i) =
3 + 4i
Hermitian transpose of v = 3+4i - write syntax
np.transpose(v.conjugate()) * v
Unit vectors - mu =
1/np.linalg.norm(v)
Normalize a vector
mu * v= norm_vector
np.linalg.norm(norm_vector) = 1
Scalar multiplication
2 * v
Creating an identity matrix
np.eye(3), np.eye(3,5); But identity matrices are only square matrices in practice
Creating zeros matrix
np.zeros((3,3)), np.zeros((3,5))
creating diagonal matrix
np.diag([1,2,3,4,5]); Can create rectangular matrices as well
Creating upper triangular matrix
s = np.random.randn(5,5)
ut = np.triu(s); Can create rectangular matrices as well
Creating lower triangular matrix
s = np.random.randn(5,5)
tl = np.tril(s); Can create rectangular matrices as well
Concatenate two matrices
np.concatenate((A,B), axis = 1)
Addition of matrices
A + B; Both the matrices must be of same size
Shifting a matrix
d = np.random.randn(3,3)
si = 0.3 * eye(3)
shiftedD = d + si
Create 2x2 matrix
np.array([[2,3], [3,4]])
Convert np array to matrix
c = np.array([1, 2+9j, 3, 4])
m = np.matrix(c)
Transposes on matrix
- np.matrix(c).H
- np.matrix(c).T
Two applications of diag keyword
- d = np.diag(matrix) - creates a row vector with the diagonal elements as row vector
- x = np.diag(d) - creates a diagonal matrix with the elements of the row vector as diagonal elements.
Calculate trace
- np.trace(matrix)
- np.sum(np.diag(matrix)))
- sum(np.diag(matrix)))
Properties that prove trace is a linear operator
- np.trace(a+b) = np.trace(a) + np.trace(b)
- l * np.trace(a) = np.trace(l*a)
closed under both addition and subtraction
Create a matrix with values from 1 to 12 and shape 3X4
a = np.reshape(np.arange(1,13), (3,4), ‘F’)
Third parameter in np.reshape
F - column
C - row - default
Broadcasting row vectors
m = np.reshape(np.nrange(1,13), (3,4), ‘F’)
a = np.array([10, 20, 30, 40])
m+a
Broadcasting column vectors
a + r
a+ c
Creating a symmetric matrix from a non-symmetric matrix a
a * a.T/m ** 2
Create a random integer matrix
np.random.randint(1,12, (m,m))
Hadamard matrix multiplication
- np.multiply( a, b)
- a*b
a@b - regular matrix multiplication
Hadamard multiplication rule
Both the matrices must be of same size
