VOOZH about

URL: https://www.geeksforgeeks.org/numpy/numpy-linear-algebra/

⇱ Numpy | Linear Algebra - GeeksforGeeks

  • Courses
  • Tutorials
  • Interview Prep

Numpy | Linear Algebra

Last Updated : 20 May, 2026

NumPy provides a linalg module for performing linear algebra operations on arrays and matrices. It includes functions for finding matrix rank, determinant, trace, eigenvalues, matrix inversion, matrix exponentiation and solving linear equations.

Output:

Rank of A: 3
Trace of A: 11
Determinant of A: -306.0
Inverse of A:
[[ 0.17647059 -0.00326797 -0.02287582]
[ 0.05882353 -0.13071895 0.08496732]
[-0.11764706 0.1503268 0.05228758]]
Matrix A raised to power 3:
[[336 162 228]
[406 162 469]
[698 702 905]]

Matrix and Vector Products

1. Using dot() function: returns the dot product of two arrays. It works with both 1D (vectors) and 2D (matrices). When used on 2D arrays, it performs matrix multiplication.

For N dimensions it is a sum product over the last axis of a and the second-to-last of b:

dot(a, b)[i, j, k, m] = sum(a[i, j, :] * b[k, :, m])


Output
Dot Product of scalar values: 20
Dot Product: (-7+22j)

Explanation:

a = 2 + 3j
b = 4 + 5j

Now, dot product
= 2(4 + 5j) + 3j(4 - 5j)
= 8 + 10j + 12j - 15
= -7 + 22j

2. Using vdot() function: returns the dot product, but it takes the complex conjugate of the first argument before multiplying. This makes it useful for complex-valued arrays.


Output
Dot Product: (23-2j)

Explanation:

a = 2 + 3j
b = 4 + 5j

As per method, take conjugate of a i.e. 2 - 3j
Now, dot product = 2(4 - 5j) + 3j(4 - 5j)
= 8 - 10j + 12j + 15
= 23 - 2j

Common Matrix and Vector Product Functions

Following table lists commonly used NumPy functions for performing various types of matrix and vector multiplications:

FUNCTIONDESCRIPTION
matmul()Matrix product of two arrays.
inner()Inner product of two arrays.
outer()Compute the outer product of two vectors.
linalg.multi_dot()Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order.
tensordot()Compute tensor dot product along specified axes for arrays >= 1-D.
einsum()Evaluates the Einstein summation convention on the operands.
einsum_path()Evaluates the lowest cost contraction order for an einsum expression by considering the creation of intermediate arrays.
linalg.matrix_power()Raise a square matrix to the (integer) power n.
kron()Kronecker product of two arrays.

Solving Equations and Inverting Matrices

1. Using linalg.solve() function: It is used to find the exact solution of a linear system of equations of the form Ax = b, where:

  • A is the coefficient matrix
  • b is the constant matrix
  • x is the unknown we want to solve for

Output
('Solution of linear equations:', array([2., 3.]))

2. Using linalg.lstsq() function: finds the best possible fit minimizing the difference between When a system doesn’t have an exact solution (for example, when it’s overdetermined), predicted and actual values.

Output

👁 4

Common Functions for Solving and Inverting Matrices

Below are some of the most useful functions to solve, invert or compute pseudoinverses of matrices:

FUNCTIONDESCRIPTION
numpy.linalg.tensorsolve()Solve the tensor equation a x = b for x.
numpy.linalg.inv()Compute the (multiplicative) inverse of a matrix.
numpy.linalg.pinv()Compute the (Moore-Penrose) pseudo-inverse of a matrix.
numpy.linalg.tensorinv()Compute the ‘inverse’ of an N-dimensional array.

Special Functions in Linear Algebra

1. Finding the Determinant - numpy.linalg.det(): The determinant is a number that can be calculated from a square matrix. It helps determine whether a matrix is invertible and is often used in solving systems of linear equations.


Output
('Determinant of A:', np.float64(-306.0))

2. Finding the Trace - numpy.trace(): The trace of a matrix is the sum of its diagonal elements. It’s often used in linear algebra and statistics.


Output
Trace of A: 11

Common Special Functions

Here’s a list of other specialized linear algebra functions available in NumPy:

FUNCTIONDESCRIPTION
numpy.linalg.norm()Matrix or vector norm.
numpy.linalg.cond()Compute the condition number of a matrix.
numpy.linalg.matrix_rank()Return matrix rank of array using SVD method
numpy.linalg.cholesky()Cholesky decomposition.
numpy.linalg.qr()Compute the qr factorization of a matrix.
numpy.linalg.svd()Singular Value Decomposition.

Matrix Eigenvalues Functions

NumPy provides functions in its linalg (Linear Algebra) module to calculate eigenvalues and eigenvectors of matrices.

1. Using linalg.eigh() function: It is used for Hermitian (complex symmetric) or real symmetric matrices. It returns two values:

  • An array of eigenvalues
  • A matrix of eigenvectors (each column corresponds to one eigenvalue)

Output

Array is:
[[ 1.+0.j -0.-2.j]
[ 0.+2.j 5.+0.j]]
Eigenvalues are: [0.17157288 5.82842712]
Eigenvectors are:
[[-0.92387953+0.j -0.38268343+0.j ]
[ 0. +0.38268343j 0. -0.92387953j]]

2. Using linalg.eig() function: It is used for general square matrices (not necessarily symmetric). It also returns both eigenvalues and eigenvectors.


Output
Array is:
 [[1 0 0]
 [0 2 0]
 [0 0 3]]
Eigenvalues are: [1. 2. 3.]
Eigenvectors are:
 [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

Common Eigenvalue Functions

This table summarizes various functions related to eigenvalue and eigenvector computations:

FUNCTIONDESCRIPTION
linalg.eigvals()Compute the eigenvalues of a general matrix.
linalg.eigvalsh(a[, UPLO])Compute the eigenvalues of a complex Hermitian or real symmetric matrix.
Comment
Article Tags:
Article Tags:

Explore