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A linear endomorphism of $R^{n}$ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant

Statement

Let $n \geq 1$ and let $T : R^{n} \to R^{n}$ be linear, with standard matrix $A$ . For every bounded Jordan set $E$ , the image $T (E)$ is a bounded Jordan set and $cont (T (E)) = ∣ \det A ∣ cont (E) .$ In particular, a singular linear image has content zero.

Facts & Assumptions

Given: A linear endomorphism $T$ with matrix $A$ and a bounded Jordan set $E$ .

[L1]

For every $n \geq 1$ and $A \in M_{n} (R)$ , the matrix $A$ is invertible if and only if $\det (A) \neq 0$ (A finite square real matrix is invertible if and only if its determinant is nonzero); every invertible $A \in M_{n} (R)$ is a finite product of elementary matrices, with the identity represented by the empty product (Every invertible finite square real matrix is a finite product of elementary matrices).

[L2]

For $n \geq 1$ and $A \in M_{n} (R)$ over a commutative ring, interchanging two rows changes $\det (A)$ to $- \det (A)$ , multiplying one row by $c \in R$ changes it to $c \det (A)$ , and adding $c$ times one row to a distinct row leaves it equal to $\det (A)$ (For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged).

[L3]

Cavalieri identifies content with the integral of sectional contents (Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content).

[L4]

Lipschitz self-maps of Euclidean space preserve null sets (A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets), and a bounded set is Jordan measurable exactly when its boundary is null (A bounded set in $R^{m}$ is Jordan measurable iff its boundary is null, equivalently of content zero).

[L5]

Every Euclidean linear map is bounded and continuous (Every Euclidean linear map has a unique matrix and satisfies $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ ).

[L6]

Every finite matrix over a field is row equivalent, by Gaussian elimination, to a matrix in row echelon form (Gaussian elimination reduces every finite matrix over a field to row echelon form). For $A \in M_{n} (F)$ , the matrix $A$ is invertible if and only if it has a pivot in every row and every column (Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent).

[L7]

If $r$ elementary row operations transform $A$ into $B$ , and $E_{1}, \dots, E_{r}$ are their elementary matrices in execution order, then $B = E_{r} \dots E_{1} A$ ; for $r = 0$ the empty product is the identity and $B = A$ (A finite row reduction from $A$ to $B$ is encoded by $B = E_{r} \dots E_{1} A$ ). Every elementary matrix $E \in M_{n} (F)$ is invertible, with inverse the elementary matrix of the inverse row operation (Every elementary matrix is invertible, with inverse given by the reverse elementary operation).

[L8]

Jordan inner and outer content approximate a Jordan set by finite rectangular figures (Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ ).

Proof

technique · constructive

1.1

Suppose first that $A$ is invertible. For every elementary matrix $E_{0}$ , both $E_{0}$ and $E_{0}^{- 1}$ are Lipschitz by [L5], so $\partial (E_{0} F) = E_{0} (\partial F)$ for every bounded set $F$ . Thus [L4] makes $E_{0} F$ Jordan whenever $F$ is Jordan. Coordinate permutations and nonzero coordinate scalings send rectangular figures to rectangular figures, with content factor $1$ and $∣ c ∣$ respectively; applying this to arbitrarily close inner and outer figures from [L8] proves those factors for every bounded Jordan $F$ . For a shear adding $c$ times one coordinate to another, take rectangular figures $P \subseteq F \subseteq Q$ from [L8] with $cont (Q) - cont (P)$ arbitrarily small. Sections of a rectangular figure parallel to the changed coordinate are finite unions of intervals, hence Jordan at every parameter, and the corresponding sections of $E_{0} P$ and $E_{0} Q$ are their translates by a quantity depending only on the fixed coordinates, so they are finite unions of intervals of the same total length. Both hypotheses of [L3] are therefore met by $P, Q, E_{0} P, E_{0} Q$ but are not claimed for $F$ , whose sections need not be Jordan; [L3] gives $cont (E_{0} P) = cont (P)$ and $cont (E_{0} Q) = cont (Q)$ . Since $E_{0} P \subseteq E_{0} F \subseteq E_{0} Q$ and $E_{0} F$ is already known Jordan, its content and that of $F$ are both squeezed between $cont (P)$ and $cont (Q)$ , so a shear preserves content. These are exactly the absolute determinant factors listed by [L2].

L2L3L4L5L8construct

2.1

By [L1], write $A$ as a finite product of elementary matrices. Apply step 1.1 successively to $E$ : every intermediate image is bounded Jordan, and its content is multiplied by the corresponding absolute determinant factor. The row-operation laws [L2], applied successively from the identity, identify the product of those factors with $∣ \det A ∣$ . Boundedness follows from [L5].

L1L2L5step 1.1

3.1

If $A$ is singular, [L6] reduces it to an echelon matrix with a zero row; [L7] realizes this as an invertible change of codomain coordinates. The transformed range lies in a coordinate hyperplane, whose bounded part fits in slabs of arbitrarily small thickness and therefore has content zero and is Jordan. Applying the invertible case to undo the coordinate change shows that $T (E)$ has content zero and is Jordan; the real criterion in [L1] gives $\det A = 0$ , so the same formula holds.

L1L6L7step 2.1discharge-construct: invertible and singular constructions∎

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Sources

A. Leibman, Multidimensional Real Analysis, Lemmas 5.5.2--5.5.4 (standard reference, not scraped)

A linear endomorphism of RnRn sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant