Eigenvalue equation is one of 2 eigenvalues formulas in the linear algebra section of this library, and it is used at university level.
Why eigenvalue equation works
A transformation usually turns a vector as well as resizing it. This equation picks out the special vectors that come out lying on the very same line they started on, so the matrix acts on them like a single number. The value λ records how much that line got stretched, and a negative λ means it also got reversed.
What each symbol means
$\lambda$ is an eigenvalue and $\mathbf v$ its eigenvector.
Eigenvalue equation: when it holds
$\mathbf v\ne\mathbf0$.
When it stops applying
A real matrix need not have any real eigenvalue. The quarter-turn rotation [[0, −1], [1, 0]] moves every direction off its own line, and its eigenvalues are the complex numbers i and −i, so no real vector survives with only a stretch.
Eigenvalue equation: a worked example
For diagonal $A=\operatorname{diag}(2,3)$, $(1,0)$ has eigenvalue $2$.
The mistake to avoid
What people do: Students offer the zero vector as an eigenvector, since A times zero is certainly zero times λ.
Why it goes wrong: It satisfies the equation for every possible λ, so allowing it would make every number an eigenvalue and make the idea useless.
Do this instead: Insist that v is nonzero. For A = [[2, 0], [0, 3]] the genuine eigenvectors are the coordinate directions: (1, 0) with eigenvalue 2 and (0, 1) with eigenvalue 3.
Eigenvalue equation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\lambda$ is an eigenvalue and $\mathbf v$ its eigenvector.
- Check the conditions before substituting. $\mathbf v\ne\mathbf0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most linear algebra slips.
Where this formula fits
- Subject
- Linear Algebra formulas — 17 entries in this library
- Topic
- Eigenvalues
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where eigenvalue equation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Eigenvalues and Eigenvectors — the lesson behind this formula: find directions preserved by a linear transformation.
- Characteristic Polynomial Calculator — check your substitution and the value it produces.
- Study linear algebra — the subject guide that explains the ideas these formulas compress.
- Linear Algebra Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 linear algebra formulas — the full grouped reference, or the complete formula library.
Questions about eigenvalue equation
Can an eigenvalue be zero?
Yes, and it is informative when it happens. A zero eigenvalue means some nonzero vector gets crushed to the origin, so the matrix is singular and its determinant is 0.
Is an eigenvector unique?
No. If v works, so does every nonzero multiple of it, since scaling both sides changes nothing. What is really unique is the direction, which is why answers are often normalized to length 1.
Can two different eigenvalues share one eigenvector?
No. That would force λ₁v = λ₂v with v nonzero, which forces the two eigenvalues to be equal. Each eigenvector belongs to exactly one eigenvalue.
Why do people care about these directions?
Because along them a complicated transformation becomes plain multiplication. That turns repeated application into raising a number to a power, which is what makes population models, vibration analysis, and PageRank tractable.