Introduction to Eigenvector Calculator:
The eigenvector calculator with steps is an online tool that helps you to determine the eigenvector from the given system of linear equations. It used the linear transformation method in which it finds the eigenvalue from the given equation and then finds the eigenvectors from that scalar value.
The eigenspace calculator is a useful tool specially designed to compute the eigenvectors for the square matrix for everyone because it is used in various fields such as physics, computer science, and economics.
What is an Eigenvector?
An eigenvector is a special vector that is related to a linear transformation of a vector space because it involves the system of linear equations.
An eigenvector is defined as a square matrix A, an eigenvector v, and a corresponding eigenvalue λ that satisfies the given system of linear equations. It can be expressed as:
$$ det( A - λI) \;=\; 0 $$
$$ Av \;=\; λv $$
where:
- A is a n×n square matrix.
- v is a non-zero vector in R (for complex numbers).
- λ is a scalar known as the eigenvalue associated with the eigenvector v
How to Calculate Eigenvectors?
To calculate the eigenvectors of a matrix, the eigenvector calculator with steps follows some important steps to get the exact solution of your given system. These steps are:
Suppose the eigenvector of the given system is,
$$ A \;=\; \biggr[\begin{matrix}-5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] $$
Step-by-Step Process to Find Eigenvectors
First, the eigenspace basis calculator converts the given system of equations into an augmented matrix, then finds the determinant with the help of the value equation that used in the determinant as
$$ det( A - λI) \;=\; 0 $$
$$ det \biggr(λ \biggr[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \biggr] - \biggr[\begin{matrix} -5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] \biggr)I) \;=\; 0 $$
$$ det \biggr[\begin{matrix} λ + 5 & -2 \\ 7 & λ - 4 \\ \end{matrix} \biggr] \;=\; 0 $$
Then the eigenbasis calculator solves the determinant and you get the quadratic equation,
$$ λ^2 + λ - 6 \;=\; 0 $$
$$ λ^2 + 3λ - 2λ - 6 \;=\; 0 $$
$$ λ(\lambda + 3) - 2(\lambda + 3) \;=\; 0 $$
$$ (\lambda + 3) (\lambda - 2) \;=\; 0 $$
$$ (\lambda + 3) \;=\; 0 \;,\; (\lambda - 2) \;=\; 0 $$
$$ λ \;=\; -3 \;,\; λ \;=\; 2 $$
Now the eigenvector finder adds these scalar values into the above equation as:
For λ=2 as det(2I-λ)=0
$$ \biggr(2 \biggr[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \biggr] - \biggr[\begin{matrix} -5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] \biggr) \biggr[\begin{matrix} x \\ y \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} 0 \\ 0 \\ \end{matrix} \biggr] $$
$$ \biggr[\begin{matrix} 7 & -2 \\ 7 & -2 \\ \end{matrix} \biggr] \biggr[\begin{matrix} x \\ y \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} 0 \\ 0 \\ \end{matrix} \biggr] $$
Now the eigenvector solver solves the square matrix A and transforms it into the identity matrix.
$$ \biggr[\begin{array}{rr|rr} 7 & -2 & 0 \\ 7 & -2 & 0 \\ \end{array} \biggr] \rightarrow … \rightarrow \biggr[\begin{array}{rr|rr} 1 & -2/7 & 0 \\ 0 & 0 & 0 \\ \end{array} \biggr] $$
$$ \biggr[\begin{matrix} 2/7 s \\ s \\ \end{matrix} \biggr] \;=\; s \biggr[\begin{matrix} 2/7 \\ 1 \\ \end{matrix} \biggr] $$
Further, the eigenvector calculator with steps simplifies the matrix and gets the value of s by multiplying 7.
$$ t \biggr[\begin{matrix} 2 \\ 7 \\ \end{matrix} \biggr] $$
For λ =2 the eigenvector is,
$$ \begin{matrix} 2 \\ 7 \\ \end{matrix} $$
For λ= -3 as det(-3I-λ)=0
$$ \biggr((-3) \biggr[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \biggr] - \biggr[\begin{matrix} -5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] \biggr) \biggr[\begin{matrix} x \\ y \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} 0 \\ 0 \\ \end{matrix} \biggr] $$
$$ \biggr[\begin{matrix} 2 & -2 \\ 7 & -7 \\ \end{matrix} \biggr] \biggr[\begin{matrix} x \\ y \\ \end{matrix} \;=\; \biggr[\begin{matrix} 0 \\ 0 \\ \end{matrix} \biggr] $$
The eigenspace calculator repeats the above method that is used for λ= 2 to get the eigenvector of the given equation,
$$ \biggr[\begin{matrix} s \\ s \\ \end{matrix} \;=\; s \biggr[\begin{matrix} 1 \\ 1 \\ \end{matrix} \biggr] $$
For λ =-3 the eigenvector is
$$ \biggr[\begin{matrix} 1 \\ 1 \\ \end{matrix} \biggr] $$
For verification of whether the eigenvector is correct or not, it verifies AX = 2X,
$$ \biggr[\begin{matrix} -5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] \biggr[\begin{matrix} 2 \\ 7 \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} 4 \\ 14 \\ \end{matrix} \biggr] 2 \biggr[\begin{matrix} 2 \\ 7 \\ \end{matrix} \biggr] $$
Verify that AX = -3X for the basic eigenvector,
$$ \biggr[\begin{matrix} -5 & 2 \\ -7 & 4 \\ \end{matrix} \biggr] \biggr[\begin{matrix} 1 \\ 1 \\ \end{matrix} \biggr] \;=\; \biggr[\begin{matrix} -3 \\ -3 \\ \end{matrix} \biggr] -3 \biggr[\begin{matrix} 1 \\ 1 \\ \end{matrix} \biggr] $$
After verification, we know our eigenvectors are correct for the given system of linear equations.
How to Use Eigenvector Calculator?
The eigenspace basis calculator has a simple interface, that enables you to use it to evaluate the eigenvector from the given system of equation questions. Before adding the input for the solutions of the eigenvector problem, you must follow some simple steps. These steps are:
- Select the order of the matrix from the list for the eigenvector solution.
- Enter the matrix element as per your matrix order in the input box.
- Review your input value for the eigenvector solution before hitting the calculate button to start the calculation process
- Click on the “Calculate” button of the eigenbasis calculator to get the desired result of your given eigenvector problem.
- If you want to try out our eigenvector finder to check its accuracy in solution then must try the load example.
- Click on the “Recalculate” button to get a new page for solving more linear system questions.
Outcome of Eigenspace Calculator:
Eigenvector Calculator gives you the solution to a given problem when you add the input to it. It provides you with solutions of linear systems. It may contain as:
- Result Option
You can click on the result option and it provides you with a solution for the eigenvector.
- Possible Step
When you click on the possible steps option it provides you with the solution of the eigenvector from the given linear equation.
Advantages of Eigenspace Basis Calculator
The eigenvector solver gives you tons of advantages whenever you use it to calculate a system of linear equations and to get its solution immediately. These advantages are:
- Our eigenspace calculator saves the time and effort that you consume in solving complex eigenvector questions in a few seconds
- It is a free-of-cost tool that provides you solution of eigenvectors from the given system without paying a single penny.
- You can use this eigenbasis calculator for practice so that you get a strong hold on this concept.
- It is an adaptive tool that allows you to find the correct solution from the given system of equations for eigenvectors.
- The eigenvector calculator with steps is a trustworthy tool that provides you with accurate solutions as per your input system to calculate the eigenvector problem.
