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.
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.
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 $A=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$.
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.
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.
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 $p(t)=t^3-2t+5$.
Find the matrix of $q(x,y)=3x^2-4xy+y^2$.
Find the signature of $q(x,y)=x^2-y^2$ and name the curve $q=1$.
Diagonalize $q(x,y)=5x^2+4xy+5y^2$ and classify $q=1$.
Is $q(x,y)=x^2+4xy+y^2$ positive definite?
An operator has invariant factors $t-1$ and $(t-1)(t-2)$. Find its characteristic and minimal polynomials.
Show $I$ and $2I$ are congruent but not similar in $M_2(\mathbb{R})$.
Why is the rational form the right tool for similarity over $\mathbb{Q}$?
Course coverage. This unit is part of the complete MAT247H1 topic map. It should cover the following items before you move on:
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.
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?
How do they relate to the characteristic and minimal polynomials?
What is preserved by congruence of real symmetric matrices?
A quadratic form in three variables has signature $(2,1,0)$. What is $q=1$?