Rational Form, Bilinear & Quadratic Forms
Two closing threads. The rational canonical form classifies operators over any field, with no splitting assumption. And bilinear forms generalise the inner product by dropping positivity — which is what turns the spectral theorem into a classification of quadratic surfaces.
- Build a companion matrix and state the rational canonical form.
- Explain why it works over fields where Jordan form does not.
- Represent a bilinear form by a matrix and track congruence.
- Diagonalize a quadratic form and read off its signature.
Jordan form needs the characteristic polynomial to split, which fails over $\mathbb{R}$ and badly over $\mathbb{Q}$. The rational form needs nothing: it is built from the invariant factors, which live in the base field. Quadratic forms, meanwhile, are what an inner product becomes when positivity is dropped.
The second-derivative test in multivariable calculus is a statement about the signature of the Hessian quadratic form. Special relativity uses the indefinite form with signature $(1,3)$. Conic sections and quadric surfaces are classified entirely by signature.
Sylvester proved his law of inertia in 1852, showing the signature is a genuine invariant. Over $\mathbb{Q}$ the classification of quadratic forms needs the Hasse–Minkowski theorem and $p$-adic numbers — a striking jump in difficulty from the real case.
Companion matrices and the rational form
For a monic polynomial $p(t)=t^k+a_{k-1}t^{k-1}+\cdots+a_0$, the companion matrix $C(p)$ is the $k\times k$ matrix with $1$s on the subdiagonal and $-a_0,\dots,-a_{k-1}$ down the last column. Both its characteristic and its minimal polynomial equal $p$.
A subspace is $T$-cyclic if it is $\operatorname{span}\{v,Tv,T^2v,\dots\}$ for some $v$. On such a subspace, with that ordered basis, the matrix of $T$ is a companion matrix.
Let $T$ be an operator on a finite-dimensional space over any field $F$. Then $V$ is a direct sum of $T$-cyclic subspaces, and there is a basis in which
The monic polynomials $p_i$ — the invariant factors — are unique. Here $p_r$ is the minimal polynomial and $\prod_ip_i$ is the characteristic polynomial.
Two matrices over $F$ are similar if and only if they have the same invariant factors. Unlike the Jordan form this needs no splitting hypothesis, so it settles similarity over $\mathbb{Q}$ and $\mathbb{R}$ too.
Over $\mathbb{R}$, take the rotation $A=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$.
Bilinear forms
A bilinear form on $V$ over $F$ is a map $B:V\times V\to F$ linear in each argument separately. It is symmetric if $B(u,v)=B(v,u)$, and skew-symmetric if $B(u,v)=-B(v,u)$.
A real inner product is a symmetric bilinear form that is also positive definite. Dropping positivity is exactly the generalisation here; over $\mathbb{C}$ one usually keeps conjugation and studies sesquilinear forms instead.
Fix a basis $\beta=\{v_1,\dots,v_n\}$ and let $A_{ij}=B(v_i,v_j)$. Then $B(u,v)=[u]_\beta^t\,A\,[v]_\beta$. The form is symmetric iff $A^t=A$.
If $Q$ is the change-of-basis matrix then the new matrix is $Q^tAQ$. Two matrices related this way are called congruent.
This is not similarity: similarity uses $Q^{-1}AQ$. The two coincide only when $Q$ is orthogonal, which is why the spectral theorem is so useful here — it lets one change of basis do both jobs at once.
Quadratic forms and signature
The quadratic form attached to a symmetric bilinear form is $q(v)=B(v,v)$. Over a field where $2$ is invertible, $B$ is recovered by polarization: $B(u,v)=\tfrac12\left(q(u+v)-q(u)-q(v)\right)$, so forms and quadratic forms are interchangeable data.
Every real quadratic form can be diagonalized by an orthogonal change of basis: there is an orthogonal $Q$ with $Q^tAQ=D$ diagonal, and then $q(v)=\sum_i\lambda_iy_i^2$ in the new coordinates $y=Q^tv$.
This is the real spectral theorem: $A$ is symmetric, so it is orthogonally diagonalizable, and for orthogonal $Q$ congruence and similarity agree.
Any real symmetric $A$ is congruent to a diagonal matrix with entries $1$, $-1$ and $0$ only. The counts $(n_+,n_-,n_0)$ — the signature — do not depend on the diagonalization chosen.
Rescaling each nonzero diagonal entry by $1/\sqrt{|\lambda_i|}$ reduces it to $\pm1$; invariance holds because $n_+$ is the largest dimension of a subspace on which $q$ is positive definite, a description that never mentions a basis.
Classify $q(x,y)=2x^2+2xy+2y^2=1$.
A real symmetric $A$ is positive definite iff all eigenvalues are positive, iff the signature is $(n,0,0)$, iff every leading principal minor is positive (Sylvester's criterion). This is the second-derivative test: the Hessian's signature decides minimum, maximum, or saddle.
• Confusing congruence $Q^tAQ$ with similarity $Q^{-1}AQ$. They agree
only for orthogonal $Q$.
• Forgetting the halves. In $q(x,y)=ax^2+bxy+cy^2$ the matrix has $b/2$
off the diagonal, not $b$.
• Thinking eigenvalues are congruence invariants. Only their
signs are — that is the content of Sylvester's law.
• Expecting rational form to be easy to read. It is a complete invariant
but far less transparent than Jordan form; use it when the field forces
you.
Practice
Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.
Write $C(p)$ for $p(t)=t^3-2t+5$.
Answer
Find the matrix of $q(x,y)=3x^2-4xy+y^2$.
Answer
Find the signature of $q(x,y)=x^2-y^2$ and name the curve $q=1$.
Answer
Diagonalize $q(x,y)=5x^2+4xy+5y^2$ and classify $q=1$.
Solution
Is $q(x,y)=x^2+4xy+y^2$ positive definite?
Solution
An operator has invariant factors $t-1$ and $(t-1)(t-2)$. Find its characteristic and minimal polynomials.
Solution
Show $I$ and $2I$ are congruent but not similar in $M_2(\mathbb{R})$.
Solution
Why is the rational form the right tool for similarity over $\mathbb{Q}$?
Solution
Expanded topic checklist
Course coverage. This unit is part of the complete MAT247H1 topic map. It should cover the following items before you move on:
- Companion matrices and cyclic subspaces. Build $C(p)$ and recognise a $T$-cyclic subspace.
- Rational canonical form. State it with the divisibility chain and use invariant factors to test similarity.
- Bilinear forms. Represent by a matrix and track congruence under change of basis.
- Quadratic forms. Diagonalize by principal axes, compute the signature, and apply Sylvester's law.
Use the current UofT MAT247H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
Cyclic subspaces give companion blocks and the rational form; symmetric bilinear forms give quadratic forms; the real spectral theorem diagonalizes them; and Sylvester's law says the signature is all that survives.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
What advantage does it have over Jordan form?
Guide
How do they relate to the characteristic and minimal polynomials?
Guide
What is preserved by congruence of real symmetric matrices?
Guide
A quadratic form in three variables has signature $(2,1,0)$. What is $q=1$?