Lecture 10 — Reduction to Canonical Form by Orthogonal Transformation

Lecture 10 — Reduction of a Quadratic Form to Canonical Form by Orthogonal Transformation

1. Canonical form & orthogonal transformation — statement

Let \(Q(x)=x^T A x\) be a quadratic form with real symmetric matrix \(A\). Then by the spectral theorem there exists an orthogonal matrix \(P\) (i.e. \(P^T P=I\)) such that

\[ P^T A P = D = \operatorname{diag}(\lambda_1,\dots,\lambda_n), \]

and under the orthonormal change of variables \(x=Py\) we have the canonical form (principal axes):

\[ Q(x)=y^T D y = \lambda_1 y_1^2 + \dots + \lambda_n y_n^2. \]

This representation is called the canonical form of the quadratic form under orthogonal transformations. The \(\lambda_i\) are real and are the eigenvalues of \(A\).

2. Algorithm (step-by-step)

  1. Given a quadratic form, write its symmetric matrix \(A\) (split cross-terms equally between symmetric off-diagonals).
  2. Compute eigenvalues \(\lambda_i\) and an orthonormal set of eigenvectors \(u_i\) (use Gram–Schmidt if necessary).
  3. Form the orthogonal matrix \(P=[u_1\ u_2\ \dots\ u_n]\) with eigenvectors as columns.
  4. Compute \(D=P^T A P\) which will be diagonal with \(\lambda_i\) on the diagonal.
  5. Make the change of variables \(x=Py\) to write \(Q(x)=\sum \lambda_i y_i^2\) — this is the canonical form by orthogonal transformation.

Remarks: Orthogonal transformations preserve lengths and angles, so this reduction preserves geometric features; it is numerically stable compared to arbitrary similarity transformations.

3. Worked examples (detailed)

Example 1 — 2×2 (full arithmetic)

Reduce \(Q(x,y)=5x^2-6xy+5y^2\) to canonical form by orthogonal transformation and classify.

Solution

Matrix: \(A=\begin{pmatrix}5 & -3\\-3 & 5\end{pmatrix}\) (since cross-term -6xy -> -3 above and below diagonal).

Compute eigenvalues: characteristic polynomial:

\[\det(A-\lambda I)=(5-\lambda)^2-9=\lambda^2-10\lambda+16.\]

Solve \(\lambda^2-10\lambda+16=0\): \(\lambda=\frac{10\pm\sqrt{100-64}}{2}=\frac{10\pm6}{2}\) so \(\lambda_1=8,\ \lambda_2=2\).

Eigenvectors: for \(\lambda=8\), solve \((A-8I)v=0\): \(\begin{pmatrix}-3&-3\\-3&-3\end{pmatrix}v=0\) so take \(v_1=(1,-1)^T\). For \(\lambda=2\), \(v_2=(1,1)^T\).

Normalize: \(u_1=\frac{1}{\sqrt{2}}(1,-1)^T,\ u_2=\frac{1}{\sqrt{2}}(1,1)^T\). Take \(P=[u_1\ u_2]\).

Then \(D=P^T A P=\operatorname{diag}(8,2)\). Change variables \(x=Py\) gives canonical form \(Q=8y_1^2+2y_2^2\); classification: positive definite (both eigenvalues positive).

Example 2 — 3×3 (distinct eigenvalues)

Reduce \(Q(x)=2x_1^2 + 2x_2^2 + 2x_3^2 + 2x_1x_2 - 4x_1x_3\).

Solution

Matrix form (symmetric):

\[ A=\begin{pmatrix}2 & 1 & -2\\1 & 2 & 0\\-2 & 0 & 2\end{pmatrix}. \]

We compute eigenvalues by solving \(\det(A-\lambda I)=0\). For brevity we show steps:

  1. Compute characteristic polynomial (expand or use software) — it factors as \((\lambda-4)(\lambda-2)^2\).
  2. Eigenvalues: \(4,2,2\) (with multiplicity two for 2).
  3. Eigenvector for \(\lambda=4\): solve \((A-4I)v=0\) gives \(v=(1,0,1)^T\) (up to scale).
  4. For \(\lambda=2\), find two independent eigenvectors: e.g. \((1,1,0)^T\) and \((1,-1,1)^T\) (verify by substitution). Orthogonalize if necessary and normalize.

Form orthonormal \(u_i\) and set \(P=[u_1\ u_2\ u_3]\). Then \(P^TAP=\operatorname{diag}(4,2,2)\) and canonical form is \(4y_1^2+2y_2^2+2y_3^2\) — positive definite.

Example 3 — Repeated eigenvalue & geometric multiplicity

Discuss reduction when eigenvalues repeat (choose orthonormal basis of eigenspace).

Solution

If an eigenvalue has multiplicity >1, its eigenspace has dimension equal to its geometric multiplicity. Any orthonormal basis of that eigenspace may be chosen as corresponding columns of \(P\). Orthogonality between different eigenspaces is automatic for symmetric matrices.

Example 2.1 — \(Q(x,y)=3x^2+2xy+3y^2\)

Solution (steps)

Matrix: \(A=\begin{pmatrix}3&1\\1&3\end{pmatrix}\).

Characteristic polynomial: \((3-\lambda)^2-1=\lambda^2-6\lambda+8\). Roots: \(\lambda=2,4\).

Eigenvectors: for \(\lambda=4\): \((A-4I)v=0\Rightarrow(-1,1)^T\). For \(\lambda=2\): \((1,1)^T\).

Orthonormal basis: \(u_1=\frac{1}{\sqrt{2}}( -1,1)^T,\ u_2=\frac{1}{\sqrt{2}}(1,1)^T\). Canonical form: \(4y_1^2+2y_2^2\) — positive definite.

Example 2.2 — \(Q(x,y)=x^2-4xy+y^2\)

Solution

Matrix: \(A=\begin{pmatrix}1&-2\\-2&1\end{pmatrix}\). Char poly: \((1-\lambda)^2-4=\lambda^2-2\lambda-3=(\lambda-3)(\lambda+1)\). Eigenvalues: 3 and -1.

Eigenvectors: for 3: (1,-1)^T; for -1: (1,1)^T. Canonical form: \(3y_1^2-1y_2^2\) — indefinite (saddle).

Example 2.3 — \(Q(x,y)=6x^2+5xy+6y^2\)

Solution

A=\(\begin{pmatrix}6&2.5\\2.5&6\end{pmatrix}\). Characteristic polynomial: compute \((6-\lambda)^2-6.25=\lambda^2-12\lambda+29.75\). Discriminant negative? \(144-119=25\) so eigenvalues real: \(\lambda=\frac{12\pm5}{2}=8.5,3.5\).

Both positive → positive definite. (Eigenvectors orthogonal; canonical form \(8.5 y_1^2 + 3.5 y_2^2\).)

Extra 3×3 Examples

Example 3.1 — \(A=\begin{pmatrix}2&0&0\\0&3&1\\0&1&3\end{pmatrix}\)

Solution

Observe block structure: first coord decouples with eigenvalue 2 (evec e1). The 2×2 block \(\begin{pmatrix}3&1\\1&3\end{pmatrix}\) has eigenvalues 4 and 2 with eigenvectors (1,1) and (1,-1).

Full eigenvalues: 2 (twice), 4. Choose orthonormal evecs: \(u_1=(1,0,0),\ u_2=\frac{1}{\sqrt{2}}(0,1,1),\ u_3=\frac{1}{\sqrt{2}}(0,1,-1)\). Canonical: \(4y_1^2+2y_2^2+2y_3^2\) — positive definite.

Example 3.2 — \(A=\begin{pmatrix}0&1&0\\1&0&0\\0&0&-2\end{pmatrix}\)

Solution

2×2 block \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\) has eigenvalues 1 and -1 with evecs (1,1,0) and (1,-1,0). Third coordinate eigenvalue -2.

Eigenvalues: 1,-1,-2 so canonical form: \(1y_1^2 -1 y_2^2 -2 y_3^2\) — signature (1,2,0) — indefinite.

Example 3.3 — \(A=\begin{pmatrix}4&1&1\\1&4&1\\1&1&4\end{pmatrix}\)

Solution

Symmetric matrix with constant diagonal and equal off-diagonals. Vector (1,1,1) is eigenvector with eigenvalue 6. Any vector orthogonal to (1,1,1) has eigenvalue 3 (multiplicity 2).

Canonical form: \(6y_1^2+3y_2^2+3y_3^2\) — positive definite.

4. Important remarks and numerical tips

5. Interactive calculator (orthogonal reduction)

Paste a symmetric 2×2 or 3×3 matrix (rows separated by semicolons; entries by spaces or commas). The tool computes eigenvalues, orthonormal eigenvectors, the diagonal matrix \(D\), and the canonical form \(\sum \lambda_i y_i^2\).