Matrix Calculator

1. What are Eigenvalues?

For a square matrix $A$, a scalar $\lambda$ is an eigenvalue if there exists a nonzero vector $\mathbf{v}$ such that

Eigenvalues measure the scaling factor by which $A$ stretches or compresses the direction $\mathbf{v}$.

2. How to compute eigenvalues (formula)

Compute eigenvalues by solving the characteristic equation:

Steps:

1. Form $A-\lambda I$.
2. Compute the determinant to get a polynomial in $\lambda$ (the characteristic polynomial).
3. Solve the polynomial equation for $\lambda$.

If $A$ is triangular (upper- or lower-triangular), its eigenvalues are simply the diagonal entries.

3. Worked examples

Example 1 (2×2)

Find eigenvalues of

Compute the characteristic polynomial:

Solve:

Example 2 (3×3, triangular)

Given an upper-triangular matrix

Eigenvalues are the diagonal entries:

4. Common mistakes

• Turning the determinant into a number: The determinant of $A-\lambda I$ must remain a polynomial in $\lambda$ until you solve for roots.
• Confusing eigenvalues with eigenvectors: Eigenvalues are scalars; eigenvectors are vectors that satisfy $A\mathbf{v}=\lambda\mathbf{v}$.
• Forgetting multiplicities: Repeated roots mean repeated eigenvalues; check algebraic vs geometric multiplicity when needed.
• Not using triangularity: For triangular matrices, read eigenvalues from the diagonal—no determinant needed.

5. Practice problems

Try these problems. Click "Show Answer" to reveal the eigenvalues.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Eigenvalues also connect naturally with the matrix inverse, matrix rank, and SVD calculator.