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7.5: Upper Triangular Matrices

( \newcommand{\kernel}{\mathrm{null}\,}\)

As before, let V be a complex vector space.

Let TL(V,V) and (v1,,vn) be a basis for V. Recall that we can associate a matrix M(T) Cn×n to the operator T. By Theorem 7.4.1, we know that T has at least one eigenvalue, say λC. Let v10 be an eigenvector corresponding to λ. By the Basis Extension Theorem, we can extend the list (v1) to a basis of V. Since Tv1=λv1, the first column of M(T) with respect to this basis is

[λ00].

What we will show next is that we can find a basis of Vsuch that the matrix M(T) is upper triangular.

Definition 7.5.1: Upper Trianglar Matrix

A matrix A=(aij)Fn×n is called upper triangular if aij=0 for i>j.

Schematically, an upper triangular matrix has the form

[0],

where the entries can be anything and every entry below the main diagonal is zero.

Here are two reasons why having an operator T represented by an upper triangular matrix can be quite convenient:

  1. the eigenvalues are on the diagonal (as we will see later);
  2. it is easy to solve the corresponding system of linear equations by back substitution (as discussed in Section A.3).

The next proposition tells us what upper triangularity means in terms of linear operators and invariant subspaces.

Proposition 7.5.2

Suppose TL(V,V) and that (v1,,vn) is a basis of V. Then the following statements are equivalent:

  1. the matrix M(T) with respect to the basis (v1,,vn) is upper triangular;
  2. Tvkspan(v1,,vk) for each k=1,2,,n;
  3. span(v1,,vk) is invariant under T for each k=1,2,,n.

Proof

The equivalence of Condition~1 and Condition~2 follows easily from the definition since Condition~2 implies that the matrix elements below the diagonal are zero.

Obviously, Condition~3 implies Condition~2. To show that Condition~2 implies Condition~3, note that any vector vspan(v1,,vk) can be written as v=a1v1++akvk. Applying T, we obtain

Tv=a1Tv1++akTvkspan(v1,,vk)

since, by Condition~2, each Tvjspan(v1,,vj)span(v1,,vk) for j=1,2,,k and since the span is a subspace of V.

The next theorem shows that complex vector spaces indeed have some basis for which the matrix of a given operator is upper triangular.

Theorem 7.5.3

Let V be a finite-dimensional vector space over C and TL(V,V). Then there exists a basis B for V such that M(T) is upper triangular with respect to B.

Proof

We proceed by induction on dim(V). If dim(V)=1, then there is nothing to prove.

Hence, assume that dim(V)=n>1and that we have proven the result of the theorem for all TL(W,W), where W is a complex vector space with dim(W)n1. By Theorem7.4.1, T has at least one eigenvalue λ.

Define

U=range(TλI),

and note that

  1. dim(U)<dim(V)=nsince λis an eigenvalue of T and hence TλI is not surjective;
  2. U is an invariant subspace of Tsince, for all uU, we have

Tu=(TλI)u+λu,

which implies that TuU since (TλI)urange(TλI)=U and λuU.

Therefore, we may consider the operator S=T|U, which is the operator obtained by restricting T to the subspace U. By the induction hypothesis, there exists a basis (u1,,um) of U with mn1 such that M(S) is upper triangular with respect to (u1,,um). This means that

Tuj=Sujspan(u1,,uj),for all j=1,2,,m.

Extend this to a basis (u1,,um,v1,,vk)of V. Then

Tvj=(TλI)vj+λvj,for all j=1,2,,k.

Since (TλI)vjrange(TλI)=U=span(u1,,um), we have that

Tvjspan(u1,,um,v1,,vj),for all j=1,2,,k.

Hence, T is upper triangular with respect to the basis (u1,,um,v1,,vk).

The following are two very important facts about upper triangular matrices and their associated operators.

Proposition 7.5.4

Suppose TL(V,V) is a linear operator and that M(T) is upper triangular with respect to some basis of V.

  1. T is invertible if and only if all entries on the diagonal of M(T) are nonzero.
  2. The eigenvalues of T are precisely the diagonal elements of M(T).

Proof of Proposition 7.5.4, Part 1

Let (v1,,vn)be a basis of V such that

M(T)=[λ10λn]

is upper triangular. The claim is that T is invertible if and only if λk0 for all k=1,2,,n. Equivalently, this can be reformulated as follows: T is not invertible if and only if λk=0 for at least one k{1,2,,n}.

Suppose λk=0. We will show that this implies the non-invertibility of T. If k=1, this is obvious since then Tv1=0, which implies that v1null(T) so that T is not injective and hence not invertible. So assume that k>1. Then

Tvjspan(v1,,vk1), for all jk,

since T is upper triangular and λk=0. Hence, we may define S=T|span(v1,,vk) to be the restriction of T to the subspace span(v1,,vk) so that

S:span(v1,,vk)span(v1,,vk1).

The linear map S is not injective since the dimension of the domain is larger than the dimension of its codomain, i.e.,

dim(span(v1,,vk))=k>k1=dim(span(v1,,vk1)).

Hence, there exists a vector 0vspan(v1,,vk) such that Sv=Tv=0. This implies that Tis also not injective and therefore also not invertible.

Now suppose that T is not invertible. We need to show that at least one λk=0. The linear map T not being invertible implies that T is not injective. Hence, there exists a vector 0vV such that Tv=0, and we can write

v=a1v1++akvk
for some k, where ak0. Then
\boldsymbol{\begin{equation}\label{eq:expansion} 
0 = Tv = (a_1 Tv_1 + \cdots + a_{k-1} Tv_{k-1}) + a_k Tv_k. \label{7.5.1} 
\end{equation}}

Since T is upper triangular with respect to the basis (v1,,vn), we know that a1Tv1++ak1Tvk1span(v1,,vk1). Hence, Equation ??? shows that Tvkspan(v1,,vk1), which implies that λk=0.

Proof of Proposition 7.5.4, Part 2.

Recall that λF is an eigenvalue of T if and only if the operator TλI is not invertible. Let (v1,,vn) be a basis such that M(T) is upper triangular. Then

M(TλI)=[λ1λ0λnλ].

Hence, by Proposition 7.5.4(1), TλI is not invertible if and only if λ=λk for some k.


This page titled 7.5: Upper Triangular Matrices is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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