Canonical Form of Matrices

1. Three Senses of "Canonical Form" in This Syllabus

The phrase is used for three different things — identify which one the question wants:

Normal (canonical) form of a matrix under elementary transformations — reducing a matrix of rank r to [Iᵣ | 0 ; 0 | 0] using row and column operations (covered in the echelon/normal-form topic).
Canonical form of a quadratic form — the diagonal (sum of squares) form λ₁y₁² + λ₂y₂² + … obtained by a change of variables. This is the main exam question of this topic.
Jordan canonical form — for a non-diagonalizable matrix, a nearly diagonal form with 1s on the superdiagonal (mentioned at the end).

2. Canonical Form of a Quadratic Form — Two Routes

Route	Transformation	Result	Note
Orthogonal reduction (principal axes)	X = QY with Q orthogonal (Qᵀ = Q⁻¹), columns = normalised eigenvectors of A	λ₁y₁² + λ₂y₂² + λ₃y₃² — the eigenvalues appear as coefficients	Preserves lengths; gives the canonical form directly
Lagrange's method (completing squares)	X = PY with P non-singular (usually not orthogonal)	a sum of squares with other coefficients	Simpler algebra; rank/index/signature come out right, coefficients are not the eigenvalues

Both are legitimate; the orthogonal route is the one usually demanded ("reduce by orthogonal transformation"), because it also yields the nature from the signs of the eigenvalues.

3. Procedure — Orthogonal Reduction (step by step)

Write the symmetric matrix A of the form.
Find its eigenvalues λ₁, λ₂, λ₃ from det(A − λI) = 0.
Find the eigenvectors. For a repeated eigenvalue, choose the independent eigenvectors inside its eigenspace and, if required to be orthogonal, apply Gram–Schmidt within that eigenspace.
Normalise each eigenvector (divide by its length) to get unit vectors.
Build Q with these unit eigenvectors as columns. Then QᵀAQ = diag(λ₁, λ₂, λ₃).
Write the canonical form λ₁y₁² + λ₂y₂² + λ₃y₃², and read off rank, index, signature and the nature.

   Reduce  Q = 2x² + 2y² + 2z² + 2xy + 2yz + 2zx  to canonical form by an orthogonal
   transformation, and state its rank, index, signature and nature.

        [ 2  1  1 ]
   A =  [ 1  2  1 ]        eigenvalues  λ = 4, 1, 1     (from the Eigenvalues topic)
        [ 1  1  2 ]

   Eigenvectors (already mutually orthogonal — see the Diagonalization topic):
        λ = 4 :  X₁ = (1, 1, 1)ᵀ              length √3      → u₁ = (1,1,1)/√3
        λ = 1 :  X₂ = (1, -1, 0)ᵀ             length √2      → u₂ = (1,-1,0)/√2
        λ = 1 :  X₃ = (1, 1, -2)ᵀ             length √6      → u₃ = (1,1,-2)/√6

        Q = [ 1/√3   1/√2    1/√6 ]
            [ 1/√3  -1/√2    1/√6 ]        Qᵀ = Q⁻¹ ,   QᵀAQ = diag(4, 1, 1)   ✓
            [ 1/√3    0    -2/√6 ]

   ⇒  CANONICAL FORM :   4y₁² + y₂² + y₃²
   ⇒  rank r = 3  (all three coefficients non-zero)
      index p = 3  (all coefficients positive)
      signature s = 2p - r = 3
      NATURE : POSITIVE DEFINITE

7. Invariants and Uniqueness

Rank, index and signature are invariant under any non-singular change of variables — so whichever route (orthogonal or Lagrange) is used, these three numbers must agree.
The canonical form itself is not unique: Lagrange's method may give different coefficients (and even different numbers of variables) than orthogonal reduction, but the signs and count of the squares always match (Sylvester's law of inertia).
The number of positive squares = number of positive eigenvalues = index; number of negative squares = index − (2·index − rank) … simply: negatives = rank − index.
If all eigenvalues are positive the form is definite and the canonical form is a sum of squares with all-plus signs.

8. Jordan Canonical Form (brief, for completeness)

A matrix that cannot be diagonalized (a defective eigenvalue) can still be reduced by similarity to its Jordan canonical form: a block-diagonal matrix whose blocks are

      [ λ  1  0 ]                                    [ λ  0  0 ]
      [ 0  λ  1 ]   for a defective λ ,   or simply  [ 0  λ  0 ]   when diagonalizable
      [ 0  0  λ ]                                    [ 0  0  λ ]

Each Jordan block carries one eigenvalue on the diagonal and 1s on the superdiagonal; the number and size of the blocks encode the geometric multiplicities. Diagonalization is the special case where every block is 1 × 1. (This is beyond the usual Maths-I question, but is the standard follow-up in Linear Algebra courses.)

9. Exam Points

Reduce a given quadratic form to canonical form by an orthogonal transformation, and state rank, index, signature and nature — 7 marks, the single most frequent question of this chapter.
Reduce a quadratic form to canonical form by Lagrange's method (completing squares) — 5-7 marks, frequent.
Define rank, index and signature; show they are invariants (Sylvester's law of inertia) — 5 marks.
State and use the normal form [Iᵣ | 0 ; 0 | 0] of a matrix — 5 marks.
Distinguish the canonical form of a quadratic form from the normal form of a matrix — 3-5 marks.
Find the nature of a form from the signs of the canonical-form coefficients — 3 marks.

Canonical Form of Matrices