Diagonalization of Square Matrices

1. Definition

A square matrix A of order n is diagonalizable if there exists a non-singular matrix P such that P⁻¹AP is a diagonal matrix D. Since A and D are similar, they share determinant, trace, rank and eigenvalues — and D, being diagonal, carries the eigenvalues on its diagonal:

P^{-1} A P = D = diag(λ_1, λ_2, ..., λ_n)

where the columns of P are the eigenvectors of A. The order matters: the eigenvector put in the i-th column of P must correspond to the eigenvalue placed at the i-th diagonal position of D.

2. The Diagonalizability Theorem

A is diagonalizable ⇔ A has n linearly independent eigenvectors ⇔ for every eigenvalue, geometric multiplicity = algebraic multiplicity.

Consequences worth quoting in an answer:

If an n × n matrix has n distinct eigenvalues, it is automatically diagonalizable (distinct eigenvalues give independent eigenvectors).
Every real symmetric matrix is diagonalizable — in fact orthogonally diagonalizable (Q⁻¹ = Qᵀ).
If for some repeated eigenvalue the number of independent eigenvectors, n − ρ(A − λI), is less than its algebraic multiplicity, the matrix is defective and cannot be diagonalized.

3. Procedure

Find all eigenvalues from det(A − λI) = 0 (with multiplicities).
For each eigenvalue find the eigenvectors by solving (A − λI)X = 0.
Count the independent eigenvectors. If the total is n → proceed; if less → stop and report "not diagonalizable (defective)".
Form P with those n eigenvectors as columns.
Write D = diag(λ₁, …, λₙ) in the matching order.
Verify (at least in principle) that det P ≠ 0 and AP = PD.

        [ 4  1 ]
   A =  [ 2  3 ]        λ = 5, 2  (distinct ⇒ diagonalizable)

   Eigenvectors:   λ = 5  →  X₁ = (1, 1)ᵀ        λ = 2  →  X₂ = (1, -2)ᵀ

        P = [ 1   1 ]   ,   D = [ 5  0 ]   ,   det P = -3 ≠ 0 ✓
            [ 1  -2 ]           [ 0  2 ]

   Verification (carried out in full in the Similarity topic):
        A·P = [ 5   2 ]      and      P⁻¹(A·P) = [ 5  0 ] = D    ✓
              [ 5  -4 ]                           [ 0  2 ]

8. Exam Points

Define a diagonalizable matrix and give the condition in terms of eigenvectors / multiplicities — 5 marks, almost certain.
Diagonalize a given 2 × 2 or 3 × 3 matrix, showing P, P⁻¹ and D — 7 marks, the most frequent question of this chapter.
Show that a matrix with n distinct eigenvalues is diagonalizable — 5 marks.
Show that a real symmetric matrix is orthogonally diagonalizable (and find the orthogonal matrix Q) — 5-7 marks, frequent.
Show that a given matrix is NOT diagonalizable, giving the reason (defective eigenvalue) — 5 marks.
Use diagonalization to compute Aᵏ (or A⁵, A⁶) — 5-7 marks, frequent.
State the relations D = P⁻¹AP and Aᵏ = PDᵏP⁻¹ and explain why they help — 3 marks.

Diagonalization of Square Matrices