Last updated on August 5th, 2025
An eigenvalue calculator is a tool designed to perform advanced linear algebra operations, specifically to find eigenvalues of matrices. It is especially helpful for completing mathematical school projects or exploring complex mathematical concepts in linear algebra. In this topic, we will discuss the Eigenvalue Calculator.
The Eigenvalue Calculator is a tool designed for calculating the eigenvalues of a matrix. In linear algebra, an eigenvalue is a scalar that indicates how much a corresponding eigenvector is stretched during a linear transformation represented by a matrix. T
he concept of eigenvalues is critical in various areas of mathematics and engineering, as they reveal important properties of a matrix. The word "eigenvalue" comes from the German word "eigen," meaning "own" or "characteristic."
For calculating the eigenvalues of a matrix using the calculator, we need to follow the steps below -
Step 1: Input: Enter the matrix elements.
Step 2: Click: Calculate Eigenvalues. By doing so, the matrix will be processed.
Step 3: You will see the eigenvalues of the matrix in the output column.
Mentioned below are some tips to help you get the right answer using the Eigenvalue Calculator.
Know the concept: The eigenvalue is found as the roots of the characteristic polynomial of a matrix.
Use the Right Dimensions: Make sure the matrix is square (same number of rows and columns) as eigenvalues are only defined for square matrices.
Enter Correct Numbers: When entering matrix elements, make sure the numbers are accurate.
Small mistakes can lead to significantly different eigenvalues, especially for large matrices.
Calculators mostly help us with quick solutions. For calculating complex math questions, students must know the intricate features of a calculator. Given below are some common mistakes and solutions to tackle these mistakes.
Help Alice find the eigenvalues of the matrix: [2, 1; 1, 2].
The eigenvalues of the matrix are 3 and 1.
To find the eigenvalues, we solve the characteristic equation: det(A - λI) = 0, where A is the matrix and I is the identity matrix.
For the matrix [2, 1; 1, 2], the characteristic equation becomes: |2-λ, 1| |1, 2-λ| = (2-λ)(2-λ) - 1*1 = λ² - 4λ + 3 = 0.
The solutions to the equation λ² - 4λ + 3 = 0 are λ = 3 and λ = 1.
The matrix [4, 0; 0, 3] is given. What are its eigenvalues?
The eigenvalues are 4 and 3.
The matrix is already diagonal, so the eigenvalues are simply the diagonal elements: 4 and 3.
Find the eigenvalues of the matrix [5, 2; 2, 5].
The eigenvalues are 7 and 3.
For the matrix [5, 2; 2, 5], the characteristic equation is: |5-λ, 2| |2, 5-λ| = (5-λ)(5-λ) - 2*2 = λ² - 10λ + 21 = 0.
The solutions to the equation λ² - 10λ + 21 = 0 are λ = 7 and λ = 3.
Determine the eigenvalues of the matrix [1, 1; 0, 1].
The eigenvalue is 1 with multiplicity 2.
For the matrix [1, 1; 0, 1], the characteristic equation is: |1-λ, 1| |0, 1-λ| = (1-λ)(1-λ) = λ² - 2λ + 1 = 0.
The solution to λ² - 2λ + 1 = 0 is λ = 1 with multiplicity 2.
Jessica needs to find the eigenvalues of the identity matrix [1, 0; 0, 1].
The eigenvalue is 1 with multiplicity 2.
The identity matrix is already diagonal, and all of its diagonal elements are 1, so the eigenvalue is 1 with multiplicity 2.
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