Eigenvalue and Eigenvector Calculator
Calculate the eigenvalues and eigenvectors of a 2x2 or 3x3 matrix.
Input Matrix (A)
Results
This is an estimate for informational purposes and does not constitute professional advice.
About the Eigenvalue & Eigenvector Calculator
This calculator is a tool for students, engineers, and data scientists working with linear algebra. It computes the eigenvalues and corresponding eigenvectors for a given 2x2 or 3x3 square matrix. These values are fundamental in understanding the properties of linear transformations, such as rotation and scaling.
Formula Explained
Eigenvalues (λ) and eigenvectors (v) are found by solving the characteristic equation of a matrix A:
- det is the determinant of the matrix.
- A is the input square matrix.
- λ (lambda) is the eigenvalue we are solving for.
- I is the identity matrix of the same size as A.
Solving this equation gives the eigenvalues. For each eigenvalue, the corresponding eigenvector is found by solving (A - λI)v = 0.
How to Interpret the Results
Eigenvalues and eigenvectors reveal the deep properties of a linear transformation:
Direction of Invariance
An eigenvector represents a direction that is unchanged by the transformation. When the matrix A is applied to its eigenvector, the vector is simply scaled.
Scaling Factor
The eigenvalue is the factor by which the eigenvector is stretched or shrunk. A value > 1 means stretching, < 1 means shrinking, and a negative value means the direction is reversed.
Frequently Asked Questions
What are eigenvalues and eigenvectors? →
In linear algebra, an eigenvector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue is the factor by which the eigenvector is scaled. Geometrically, an eigenvector corresponds to a real number λ for which there exists a nonzero vector v such that T(v) = λv.
What does an eigenvalue of 0 mean? →
An eigenvalue of 0 means that the matrix is singular, which means it does not have an inverse. Geometrically, this indicates that the linear transformation represented by the matrix collapses at least one dimension of the vector space into the zero vector. The corresponding eigenvectors form the null space (or kernel) of the matrix.
How do you find the eigenvalues of a 2x2 matrix? →
To find the eigenvalues of a 2x2 matrix [[a, b], [c, d]], you solve the characteristic equation: det(A - λI) = 0. This simplifies to the quadratic equation λ² - (a+d)λ + (ad-bc) = 0. The solutions for λ are the eigenvalues of the matrix.
What are the applications of eigenvalues and eigenvectors? →
Eigenvalues and eigenvectors have numerous applications across various fields. In physics, they are used to analyze vibrating systems and rotational dynamics. In computer science, Google's PageRank algorithm uses eigenvectors to rank web pages. They are also fundamental in machine learning for techniques like Principal Component Analysis (PCA) and in quantum mechanics to describe energy states.
Can a matrix have complex eigenvalues? →
Yes, a matrix with real-valued entries can have complex eigenvalues. This typically occurs when the linear transformation involves a rotational component. If a matrix has complex eigenvalues, they will always appear in conjugate pairs (a + bi and a - bi).