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Solve \(\det(A – \lambda I) = 0\) for \(\lambda\). Those numbers are the eigenvalues. For each eigenvalue, solve \((A – \lambda I)\mathbf{v} = \mathbf{0}\) to get an eigenvector \(\mathbf{v}\). For \(A = \begin{bmatrix}2 & 1\\1 & 2\end{bmatrix}\), the eigenvalues are 3 and 1.
How to Use the Eigenvalue Calculator
- Type the matrix one row at a time. Separate numbers with spaces and rows with semicolons. For example:
2 1; 1 2. - Press Solve. Matrices up to 4×4 are supported.
- The steps show \(A – \lambda I\), the characteristic polynomial, its roots, and an eigenvector for each eigenvalue.
What Are Eigenvalues and Eigenvectors?
Most vectors change direction when you multiply them by a matrix. An eigenvector is a special vector that only gets stretched or shrunk. The stretch factor is its eigenvalue:
\[A\mathbf{v} = \lambda\mathbf{v}.\]
Eigenvalues show up in physics (vibrations), statistics (principal component analysis), Google’s PageRank, and in solving systems of differential equations.
Worked Example: \(A = \begin{bmatrix}2 & 1\\1 & 2\end{bmatrix}\)
Step 1. \(A – \lambda I = \begin{bmatrix}2-\lambda & 1\\1 & 2-\lambda\end{bmatrix}\).
Step 2. \(\det(A – \lambda I) = (2 – \lambda)^2 – 1 = \lambda^2 – 4\lambda + 3\).
Step 3. \(\lambda^2 – 4\lambda + 3 = (\lambda – 1)(\lambda – 3) = 0\), so \(\lambda = 1\) or \(\lambda = 3\).
Step 4. For \(\lambda = 3\): \(\begin{bmatrix}-1 & 1\\1 & -1\end{bmatrix}\mathbf{v} = \mathbf{0}\) gives \(\mathbf{v} = \begin{bmatrix}1\\1\end{bmatrix}\). For \(\lambda = 1\): \(\mathbf{v} = \begin{bmatrix}-1\\1\end{bmatrix}\).
Check: \(A\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}3\\3\end{bmatrix} = 3\begin{bmatrix}1\\1\end{bmatrix}\). ✓
Quick Checks You Can Use
- The sum of the eigenvalues equals the trace (the sum of the diagonal). Above: \(3 + 1 = 2 + 2\).
- The product of the eigenvalues equals the determinant. Above: \(3 \times 1 = 2\cdot2 – 1\cdot1\).
- A triangular matrix has its eigenvalues on the diagonal.
- A symmetric matrix always has real eigenvalues.
Common Mistakes
- Subtracting λ from every entry. Only the diagonal entries change.
- Taking the zero vector as an eigenvector. Eigenvectors must be nonzero.
- Thinking an eigenvector is unique. Any nonzero multiple of an eigenvector is also an eigenvector.
Learn the Math Behind the Eigenvalue & Eigenvector Calculator
Eigenvalues are the special scale factors of a matrix, found from det(A − λI) = 0. They solve systems of differential equations.
Frequently Asked Questions
Can eigenvalues be complex?
Yes. For example, a rotation matrix has complex eigenvalues. This calculator lists the real eigenvalues and tells you when complex ones exist.
Why do my eigenvectors look different from my textbook?
Eigenvectors can be scaled by any nonzero number. \(\begin{bmatrix}1\\1\end{bmatrix}\) and \(\begin{bmatrix}2\\2\end{bmatrix}\) point the same way and are both correct.
What is the characteristic polynomial?
It is \(\det(A – \lambda I)\), a polynomial in \(\lambda\). Its roots are the eigenvalues. Use the factoring calculator or the quadratic formula calculator to solve it by hand.