Eigenvalues of a 2×2 Matrix — What They Actually Mean
Multiplying a matrix by a vector normally changes both the vector's length and its direction. For a few special vectors it changes only the length — the direction survives untouched. Those vectors are the eigenvectors, and the stretching factor for each one is its eigenvalue. That single idea is behind molecular orbital energies, vibrational modes, moments of inertia and the whole matrix formulation of quantum chemistry. This guide does the 2×2 case completely, by hand.
The defining equation
Rearranged: (A − λI) v = 0, which has a non-zero solution only if det(A − λI) = 0
That determinant condition is the characteristic equation. For a 2×2 matrix it simplifies to something you can write down instantly, with no determinant expansion at all.
The shortcut every student should know
Write the matrix as A = [ a b ; c d ] (top row a, b; bottom row c, d). Then:
where T = trace = a + d and D = determinant = ad − bc
so λ = [ T ± √(T² − 4D) ] / 2
Two consequences worth memorising, because they check your answer for free:
- λ₁ + λ₂ = trace (the sum of the diagonal entries)
- λ₁ × λ₂ = determinant
The quantity under the root can be rewritten as T² − 4D = (a − d)² + 4bc, which tells you at a glance whether the eigenvalues are real. For a symmetric matrix (b = c) it becomes (a − d)² + 4b², never negative — so a real symmetric matrix always has real eigenvalues. That is exactly why quantum-mechanical operators are built to be of this type: measured energies must come out real.
Worked example 1 — a matrix with two clean eigenvalues
A = [ 4 1 ; 2 3 ]
T = 4 + 3 = 7, D = (4)(3) − (1)(2) = 12 − 2 = 10
Characteristic equation: λ² − 7λ + 10 = 0 → (λ − 5)(λ − 2) = 0 →
λ = 5 and λ = 2
Check: 5 + 2 = 7 = trace ✓ and 5 × 2 = 10 = determinant ✓
Eigenvector for λ = 5. Solve (A − 5I)v = 0:
A − 5I = [ −1 1 ; 2 −2 ]. The first row gives −x + y = 0, so y = x.
Taking x = 1: v₁ = (1, 1).
Verify: A(1, 1) = (4×1 + 1×1, 2×1 + 3×1) = (5, 5) = 5 × (1, 1) ✓
Eigenvector for λ = 2. A − 2I = [ 2 1 ; 2 1 ]. The row
gives 2x + y = 0, so y = −2x. Taking x = 1: v₂ = (1, −2).
Verify: A(1, −2) = (4 − 2, 2 − 6) = (2, −4) = 2 × (1, −2) ✓
Notice that both rows of (A − λI) give the same equation. That is not a coincidence — it is the whole reason the determinant was set to zero. If your two rows give different equations, your λ is wrong.
Worked example 2 — the symmetric case, and its chemistry
A = [ 2 1 ; 1 2 ]
T = 4, D = 4 − 1 = 3 → λ² − 4λ + 3 = 0 → (λ − 3)(λ − 1) = 0 → λ = 3 and λ = 1
For λ = 3: A − 3I = [ −1 1 ; 1 −1 ] → y = x → v = (1, 1).
Check: A(1, 1) = (3, 3) ✓
For λ = 1: A − I = [ 1 1 ; 1 1 ] → y = −x → v = (1, −1).
Check: A(1, −1) = (2 − 1, 1 − 2) = (1, −1) ✓
The two eigenvectors are perpendicular — their dot product is (1)(1) + (1)(−1) = 0. Real symmetric matrices always give perpendicular eigenvectors for distinct eigenvalues.
Why a chemistry student should care. The simple Hückel treatment of ethene uses exactly this shape of matrix, with α on the diagonal and β off it. Its eigenvalues come out as E = α + β and E = α − β — the bonding and antibonding π levels — with combinations (1, 1) and (1, −1) of the two carbon 2p orbitals. Working "in-phase and out-of-phase" in MO theory is finding the eigenvectors of a 2×2 matrix.
Worked example 3 — complex eigenvalues (a rotation)
A = [ 0 −1 ; 1 0 ], the matrix that rotates the plane by 90°.
T = 0 + 0 = 0, D = (0)(0) − (−1)(1) = 1
λ² − 0λ + 1 = 0 → λ² = −1 → λ = +i and λ = −i
There is no real eigenvector, and geometrically that is obvious: a 90° rotation moves every direction, so no real vector can keep its direction. The eigenvalues being a complex conjugate pair is the algebra reporting a rotation. Sum = i + (−i) = 0 = trace ✓; product = i × (−i) = −i² = 1 = determinant ✓
Worked example 4 — a repeated eigenvalue
A = [ 3 1 ; 0 3 ]
T = 6, D = 9 → λ² − 6λ + 9 = (λ − 3)² = 0 → λ = 3, twice.
Now find the eigenvectors: A − 3I = [ 0 1 ; 0 0 ], which gives y = 0 and leaves x free. So there is only one independent eigenvector, (1, 0). A repeated eigenvalue does not guarantee two independent eigenvectors — such a matrix is called defective and cannot be diagonalised. Compare it with the identity matrix [ 1 0 ; 0 1 ], where λ = 1 twice but every vector is an eigenvector.
Reading the discriminant
| T² − 4D | Eigenvalues | Geometric meaning |
|---|---|---|
| Positive | Two distinct real values | Two independent directions, each simply stretched |
| Zero | One repeated real value | One eigenvalue; may have only one eigenvector direction |
| Negative | Complex conjugate pair | A rotation is involved; no real direction is preserved |
A few more facts that convert long questions into one-liners: the eigenvalues of a triangular matrix are its diagonal entries; a matrix is singular (det = 0) exactly when zero is one of its eigenvalues; and the eigenvalues of A² are the squares of those of A, with the same eigenvectors.
Common mistakes
- Sign error in the characteristic equation. It is λ² − (trace)λ + det, with a minus before the trace. Writing λ² + Tλ + D flips both answers' signs.
- Solving for the eigenvector with the wrong λ. Substitute one eigenvalue at a time and keep the working separate. Mixing them gives only v = 0, which is never a valid eigenvector.
- Thinking there is one correct eigenvector. Any non-zero multiple of an eigenvector is also an eigenvector; (1, 1), (2, 2) and (−3, −3) are all equally right. Normalise only if the question asks for a unit vector.
- Reporting "no eigenvalues" when the root is negative. They exist and are complex — say "no real eigenvalues".
- Assuming a repeated eigenvalue gives two eigenvectors. Example 4 shows it need not.
- Skipping the trace and determinant check. It takes five seconds and catches almost every arithmetic slip.
Where this appears in exams
| Exam / class | Typical use |
|---|---|
| CBSE/ICSE Class 12 maths | Matrices and determinants — the groundwork for this topic |
| JEE Main / Advanced | Determinant and trace properties, characteristic-equation style questions |
| Class 12 chemistry | Bonding and antibonding combinations in MO theory (the same in-phase/out-of-phase idea) |
| IIT-JAM / GATE / CSIR-NET | Hückel secular determinants, matrix mechanics, normal modes of vibration |
Always confirm the syllabus content for your paper from the current official notification.
Check your trace and determinant first. Almost every eigenvalue error is really an arithmetic error in det(A) or in the row reduction. The matrix tool computes determinants, inverses and products so you can confirm those pieces before solving λ² − Tλ + D = 0 by hand.
Open the Matrix Calculator →Class 11–12 students meeting bonding theory for the first time usually need the algebra and the chemistry taught together. ABC Chemistry runs Class 11–12 chemistry coaching at the Gurugram centre plus online classes across India — abcchemistry.in.