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Linear Algebra features available in NebulaSolver

Published: May 15, 2024

Author: The NebulaSolver Team

Introduction

At NebulaSolver.com, we are excited to introduce new features to our Linear Algebra Solver. Our app now supports additional operations such as calculating determinants, eigenvalues, and eigenvectors, as well as performing combinations of operations. These enhancements provide greater computational power and flexibility for solving complex linear algebra problems.

Enhanced Capabilities

With the new upgrades, NebulaSolver.com's Linear Algebra Solver can handle a wide range of matrix operations, making it an indispensable tool for educational purposes and professional applications in engineering, physics, mathematics, and more.

Example Test Cases

Here are some examples of the types of matrix operations our solver can now handle:

Matrix Addition

Input:

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
D = A + B

Output:

A:
[1, 2]
[3, 4]

B:
[5, 6]
[7, 8]

D:
[6, 8]
[10, 12]

Matrix Subtraction

Input:

E = [[10, 11], [12, 13]]
F = [[1, 1], [1, 1]]
G = E - F

Output:

E:
[10, 11]
[12, 13]

F:
[1, 1]
[1, 1]

G:
[9, 10]
[11, 12]

Matrix Multiplication (different dimensions)

Input:

H = [[1, 2, 3], [4, 5, 6]]
I = [[7, 8], [9, 10], [11, 12]]
J = H @ I

Output:

H:
[1, 2, 3]
[4, 5, 6]

I:
[7, 8]
[9, 10]
[11, 12]

J:
[58, 64]
[139, 154]

Matrix Transposition

Input:

K = [[1, 2], [3, 4], [5, 6]]
L = K.T

Output:

K:
[1, 2]
[3, 4]
[5, 6]

L:
[1, 3, 5]
[2, 4, 6]

Matrix Inversion

Input:

M = [[4, 7], [2, 6]]
N = M.I

Output:

M:
[4, 7]
[2, 6]

N:
[0.6000000000000001, -0.7000000000000001]
[-0.2, 0.4]

A Combined Sequence of Operations

Input:

X = [[2, -1], [0, 3]]
Y = [[8, 5], [3, 4]]
Z = X @ Y
W = Z.T + Y

Output:

W:
[21, 14]
[9, 16]

X:
[2, -1]
[0, 3]

Y:
[8, 5]
[3, 4]

Z:
[13, 6]
[9, 12]

Scalar Multiplication and Power of a Matrix

Input:

O = [[1, 2], [3, 4]]
P = 2 * O
Q = O ** 2

Output:

O:
[1, 2]
[3, 4]

P:
[2, 4]
[6, 8]

Q:
[1, 4]
[9, 16]

Determinant

Input:

A = [[1, 2], [3, 4]]
det_A = A.det

Output:

A:
[1, 2]
[3, 4]

det_A: -2.0000000000000004

Determinant

Input:

B = [[4, 7], [2, 6]]
det_B = B.det

Output:

B:
[4, 7]
[2, 6]

det_B: 10.000000000000002

Eigenvalues and Eigenvectors of a Matrix

Input:

C = [[1, 2], [3, 4]]
eig_C = C.eig

Output:

C:
[1, 2]
[3, 4]

eig_C:
{
  "eigenvalues": [-0.3722813232690143, 5.372281323269014],
  "eigenvectors":
[-0.8245648401323938, -0.4159735579192842]
[0.5657674649689923, -0.9093767091321241]
}

Eigenvalues and Eigenvectors of a Matrix

Input:

D = [[4, 7], [2, 6]]
eig_D = D.eig

Output:

D:
[4, 7]
[2, 6]

eig_D:
{
  "eigenvalues": [1.127016653792583, 8.872983346207416],
  "eigenvectors":
[-0.9251134537253817, -0.8207172853316655]
[0.3796907922722067, -0.5713345233379666]
}

Combined Operations

Input:

E = [[1, 2], [3, 4]]
F = [[4, 7], [2, 6]]
G = E + F
det_G = G.det
eig_G = G.eig

Output:

E:
[1, 2]
[3, 4]

F:
[4, 7]
[2, 6]

G:
[5, 9]
[5, 10]

det_G: 4.999999999999999

eig_G:
{
  "eigenvalues": [0.3410894683618224, 14.658910531638176],
  "eigenvectors":
[-0.888067077534101, -0.6817114719281261]
[0.45971389559164, -0.7316211239716823]
}

Combined Operations

Input:

H = [[1, 2], [2, 4]]
I = H + H.T
det_I = I.det
eig_I = I.eig

Output:

H:
[1, 2]
[2, 4]

I:
[2, 4]
[4, 8]

det_I: 0

eig_I:
{
  "eigenvalues": [0, 10],
  "eigenvectors":
[-0.8944271909999159, -0.4472135954999579]
[0.4472135954999579, -0.8944271909999159]
}

Combined Sequence of Operations

Input:

J = [[2, -1], [0, 3]]
K = [[8, 5], [3, 4]]
L = J @ K
M = L.T + K
det_M = M.det
eig_M = M.eig

Output:

J:
[2, -1]
[0, 3]

K:
[8, 5]
[3, 4]

L:
[13, 6]
[9, 12]

M:
[21, 14]
[9, 16]

det_M: 210.00000000000014

eig_M:
{
  "eigenvalues": [30, 7],
  "eigenvectors":
[0.8411784753765534, -0.7071067811865475]
[0.5407575913134987, 0.7071067811865475]
}

The Power of NebulaSolver.com

By leveraging advanced algorithms, NebulaSolver.com's Linear Algebra Solver can tackle even the most challenging matrix operations, delivering accurate results in an instant. Our tool is designed to handle any number of variables, breaking free from traditional limitations and providing a seamless experience for users.

Experience the Future of Equation Solving

Ready to simplify your complex matrix operations? Visit NebulaSolver.com and explore the full capabilities of our Linear Algebra Solver. Whether you're a student, educator, or professional, our tool is here to help you achieve accurate and efficient solutions to your mathematical problems.

For more details on how our Linear Algebra Solver works, check out our related article on Advanced Linear Algebra Solving with NebulaSolver.

Disclaimer: NebulaSolver.com is a tool designed to assist in educational and professional environments. Always verify your results for critical applications.

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