11: Canonical Forms

Given a matrix \(A\), the effect of a sequence of row-operations on \(A\) is to produce \(UA\) where \(U\) is invertible. Under this “row-equivalence” operation the best that can be achieved is the reduced row-echelon form for \(A\). If column operations are also allowed, the result is \(UAV\) where both \(U\) and \(V\) are invertible, and the best outcome under this “equivalence” operation is called the Smith canonical form of \(A\) (Theorem [thm:005369]). There are other kinds of operations on a matrix and, in many cases, there is a “canonical” best possible result.

If \(A\) is square, the most important operation of this sort is arguably “similarity” wherein \(A\) is carried to \(U^{-1}AU\) where \(U\) is invertible. In this case we say that matrices \(A\) and \(B\) are similar, and write \(A \sim B\), when \(B = U^{-1}AU\) for some invertible matrix \(U\). Under similarity the canonical matrices, called Jordan canonical matrices, are block triangular with upper triangular “Jordan” blocks on the main diagonal. In this short chapter we are going to define these Jordan blocks and prove that every matrix is similar to a Jordan canonical matrix.

Here is the key to the method. Let \(T : V \to V\) be an operator on an \(n\)-dimensional vector space \(V\), and suppose that we can find an ordered basis \(B\) of \(B\) so that the matrix \(M_B(T)\) is as simple as possible. Then, if \(B_0\) is any ordered basis of \(V\), the matrices \(M_B(T)\) and \(M_{B_0}(T)\) are similar; that is,

\[M_B(T) = P^{-1} M_{B_0}(T)P \quad \mbox{for some invertible matrix } P \nonumber \]

Moreover, \(P=P_{B_0 \leftarrow B}\) is easily computed from the bases \(B\) and \(D\) (Theorem [thm:028802]). This, combined with the invariant subspaces and direct sums studied in Section [sec:9_3], enables us to calculate the Jordan canonical form of any square matrix \(A\). Along the way we derive an explicit construction of an invertible matrix \(P\) such that \(P^{-1}AP\) is block triangular.

This technique is important in many ways. For example, if we want to diagonalize an \(n \times n\) matrix \(A\), let \(T_A : \mathbb{R}^n \to \mathbb{R}^n\) be the operator given by \(T_A(\mathbf{x}) = A\mathbf{x}\) or all \(\mathbf{x}\) in \(\mathbb{R}^n\), and look for a basis \(B\) of \(\mathbb{R}^n\) such that \(M_B(T_A)\) is diagonal. If \(B_0 = E\) is the standard basis of \(\mathbb{R}^n\), then \(M_E(T_A)=A\), so

\[P^{-1}AP = P^{-1}M_E(T_A)P = M_B(T_A) \nonumber \]

and we have diagonalized \(A\). Thus the “algebraic” problem of finding an invertible matrix \(P\) such that \(P^{-1}AP\) is diagonal is converted into the “geometric” problem of finding a basis \(B\) such that \(M_B(T_A)\) is diagonal. This change of perspective is one of the most important techniques in linear algebra.