Eigenvalues: Meaning, Characteristic Equation and a Worked Example
An eigenvalue tells you how much a matrix stretches, shrinks or flips one special direction. Here is what that means, how to find eigenvalues with the characteristic equation, and how to check your answer.
- What eigenvalues are
- Intuition
- Characteristic equation
- How to calculate
- 2×2 worked example
- Repeated eigenvalues
- Geometric meaning
- Common mistakes
What eigenvalues are
For a square matrix A, a number \lambda is an eigenvalue if some non-zero vector \mathbf v satisfies
The vector \mathbf v is an eigenvector. In words: multiplying by the whole matrix A does the same thing to \mathbf v as multiplying by the single number \lambda. An n\times n matrix has at most n eigenvalues (counted with repeats), and they can be complex numbers.
Intuition
Most vectors are both stretched and turned by a matrix. A few special directions are only scaled. The eigenvalue is that scale factor: \lambda=3 makes the vector three times longer, \lambda=0.5 halves it, \lambda=-1 flips it around, and \lambda=0 squashes it to the origin.
The characteristic equation
Start from A\mathbf v=\lambda\mathbf v and move everything to one side:
A non-zero \mathbf v can only solve this if A-\lambda I is singular, which means its determinant is zero:
This is the characteristic equation. The left side is a polynomial in \lambda of degree n, called the characteristic polynomial, and its roots are the eigenvalues. For a 2×2 matrix it simplifies to \lambda^2-\operatorname{tr}(A)\,\lambda+\det(A)=0.
How to calculate eigenvalues
- Form A-\lambda I: subtract \lambda from each diagonal entry only.
- Compute \det(A-\lambda I) to get the characteristic polynomial.
- Set it equal to zero and solve for \lambda.
- Check: the eigenvalues add up to the trace of A and multiply to \det(A).
Worked example (2×2)
Find the eigenvalues of A=\begin{bmatrix}4&1\\2&3\end{bmatrix}.
Subtract \lambda on the diagonal and take the determinant:
Factor: \lambda^2-7\lambda+10=(\lambda-5)(\lambda-2)=0, so \lambda_1=5 and \lambda_2=2.
Check: 5+2=7=\operatorname{tr}(A) and 5\cdot2=10=\det(A). ✓
Try it: enter this matrix in the Eigen Vector tool and compare the steps. Next, find the matching vectors in the eigenvectors guide.
Repeated eigenvalues
If a root appears more than once, for example (\lambda-3)^2=0, the eigenvalue is repeated. The number of times it appears is its algebraic multiplicity. Repeats can behave very differently:
- \begin{bmatrix}3&0\\0&3\end{bmatrix} has \lambda=3 twice and two independent eigenvectors.
- \begin{bmatrix}1&1\\0&1\end{bmatrix} has \lambda=1 twice but only one independent eigenvector.
The number of independent eigenvectors (the geometric multiplicity) is never larger than the algebraic multiplicity. This difference decides whether a matrix can be diagonalized.
Geometric meaning
Each real eigenvalue belongs to a line through the origin that the matrix sends to itself. If |\lambda|>1 the line is stretched, if 0<|\lambda|<1 it is compressed, a negative \lambda also flips it, \lambda=1 leaves it fixed and \lambda=0 collapses it (so \det A=0). Complex eigenvalues mean there is no such line: the matrix rotates. Also, \det A is the product of the eigenvalues, the factor by which areas change. See eigenvalues and matrix transformations.
Common mistakes
- Subtracting \lambda from every entry instead of only the diagonal.
- Writing \det(A)-\lambda instead of \det(A-\lambda I).
- Cancelling a factor of \lambda and losing the root \lambda=0.
- Skipping the trace and determinant check, which catches most slips.
- Assuming a repeated eigenvalue always gives as many eigenvectors as repeats.
FAQ
Can an eigenvalue be zero?
Yes. \lambda=0 is an eigenvalue exactly when the matrix is singular, that is, \det A=0.
Can eigenvalues be complex?
Yes. A real matrix such as a 90° rotation has the complex eigenvalues \pm i and no real eigenvectors.
How many eigenvalues does a matrix have?
An n\times n matrix has exactly n eigenvalues when counted with multiplicity and allowing complex values.