Transformations of rank one

We conclude our discussion of rank by a description of the matrices of linear transformations of rank $\leq 1$ .

Theorem 1. If a linear transformation $A$ on a finite-dimensional vector space $𝒱$ is such that $ρ (A) \leq 1$ (that is, $ρ (A) = 0$ or $ρ (A) = 1$ ), then the elements of the matrix $[A] = (α_{i j})$ of $A$ have the form $α_{i j} = β_{i} γ_{j}$ in every coordinate system; conversely if the matrix of $A$ has this form in some one coordinate system, then $ρ (A) \leq 1$ .

Proof. If $ρ (A) = 0$ , then $A = 0$ , and the statement is trivial. If $ρ (A) = 1$ , that is, $ℛ (A)$ is one-dimensional, then there exists in $ℛ (A)$ a non-zero vector $x_{0}$ (a basis in $ℛ (A)$ ) such that every vector in $ℛ (A)$ is a multiple of $x_{0}$ . Hence, for every $x$ , $A x = y_{0} x_{0},$ where the scalar coefficient $y_{0}$ ( $= y_{0} (x)$ ) depends, of course, on $x$ . The linearity of $A$ implies that $y_{0}$ is a linear functional on $𝒱$ . Let $𝒳 = {x_{1}, \dots, x_{n}}$ be a basis in $𝒱$ , and let $(α_{i j})$ be the corresponding matrix of $A$ , so that $A x_{j} = \sum_{i} α_{i j} x_{i} .$ If \mathcal{X}^{\prime}=\{y_{1}, \ldots, y_{n}\} is the dual basis in \mathcal{V}^{\prime} , then (cf. Section: Adjoints of projections , (2)) $α_{i j} = [A x_{j}, y_{i}] .$ In the present case $α_{i j} = [y_{0} (x_{j}) x_{0}, y_{i}] = y_{0} (x_{j}) [x_{0}, y_{i}] = [x_{0}, y_{i}] [x_{j}, y_{0}];$ in other words, we may take $β_{i} = [x_{0}, y_{i}]$ and $γ_{j} = [x_{j}, y_{0}]$ .

Conversely, suppose that in a fixed coordinate system $𝒳 = {x_{1}, \dots, x_{n}}$ the matrix $(α_{i j})$ of $A$ is such that $α_{i j} = β_{i} γ_{j}$ . We may find a linear functional $y_{0}$ such that $γ_{j} = [x_{j}, y_{0}]$ , and we may define a vector $x_{0}$ by $x_{0} = \sum_{k} β_{k} x_{k}$ . The linear transformation $\tilde{A}$ defined by $\tilde{A} x = y_{0} (x) x_{0}$ is clearly of rank one (unless, of course, $α_{i j} = 0$ for all $i$ and $j$ ), and its matrix $({\tilde{α}}_{i j})$ in the coordinate system $𝒳$ is given by ${\tilde{α}}_{i j} = [\tilde{A} x_{j}, y_{i}]$ (where \mathcal{X}^{\prime}=\{y_{1}, \ldots, y_{n}\} is the dual basis of $𝒳$ ). Hence ${\tilde{α}}_{i j} = [y_{0} (x_{j}) x_{0}, y_{i}] = [x_{0}, y_{i}] [x_{j}, y_{0}] = β_{i} γ_{j},$ and, since $A$ and $\tilde{A}$ have the same matrix in one coordinate system, it follows that $\tilde{A} = A$ . This concludes the proof of the theorem. ◻

The following theorem sometimes makes it possible to apply Theorem 1 to obtain results about an arbitrary linear transformation.

Theorem 2. If $A$ is a linear transformation of rank $ρ$ on a finite-dimensional vector space $𝒱$ , then $A$ may be written as the sum of $ρ$ transformations of rank one.

Proof. Since $A 𝒱 = ℛ (A)$ has dimension $ρ$ , we may find $ρ$ vectors $x_{1}, \dots, x_{ρ}$ that form a basis for $ℛ (A)$ . It follows that, for every vector $x$ in $𝒱$ , we have $A x = \sum_{i = 1}^{ρ} ξ_{i} x_{i},$ where each $ξ_{i}$ depends, of course, on $x$ ; we write $ξ_{i} = y_{i} (x)$ . It is easy to see that $y_{i}$ is a linear functional. In terms of these $y_{i}$ we define, for each $i = 1, \dots, ρ$ , a linear transformation $A_{i}$ by $A_{i} x = y_{i} (x) x_{i}$ . It follows that each $A_{i}$ has rank one and $A = \sum_{i = 1}^{ρ} A_{i}$ . (Compare this result with Section: Linear transformations , example (2).) ◻

A slight refinement of the proof just given yields the following result.

Theorem 3. Corresponding to any linear transformation $A$ on a finitedimensional vector space $𝒱$ there is an invertible linear transformation $P$ for which $P A$ is a projection.

Proof. Let $ℛ$ and $𝒩$ , respectively, be the range and the null-space of $A$ , and let ${x_{1}, \dots, x_{ρ}}$ be a basis for $ℛ$ . Let $x_{ρ + 1}, \dots, x_{n}$ be vectors such that ${x_{1}, \dots, x_{n}}$ is a basis for $𝒱$ . Since $x_{i}$ is in $ℛ$ for $i = 1, \dots, ρ$ , we may find vectors $y_{i}$ such that $A y_{i} = x_{i}$ ; finally, we choose a basis for $𝒩$ , which we may denoted by ${y_{ρ + 1}, \dots, y_{n}}$ . We assert that ${y_{1}, \dots, y_{n}}$ is a basis for $𝒱$ . We need, of course, to prove only that the $y$ ’s are linearly independent. For this purpose we suppose that $\sum_{i = 1}^{n} α_{i} y_{i} = 0$ ; then we have (remembering that for $i = ρ + 1, \dots, n$ the vector $y_{i}$ belongs to $𝒩$ ) $A (\sum_{i = 1}^{n} α_{i} y_{i}) = \sum_{i = 1}^{ρ} α_{i} x_{i} = 0,$ whence $α_{1} = \dots = α_{ρ} = 0$ . Consequently $\sum_{i = ρ + 1}^{n} α_{i} y_{i} = 0$ ; the linear independence of $y_{ρ + 1}, \dots, y_{n}$ shows that the remaining $α$ ’s must also vanish.

A linear transformation $P$ , of the kind whose existence we asserted, is now determined by the conditions $P x_{i} = y_{i}$ , $i = 1, \dots, n$ . Indeed, if $i = 1, \dots, ρ$ , then $P A y_{i} = P x_{i} = y_{i}$ , and if $i = ρ + 1, \dots, n$ , then $P A y_{i} = P 0 = 0$ . ◻

Consideration of the adjoint of $A$ , together with the reflexivity of $𝒱$ , shows that we may also find an invertible $Q$ for which $A Q$ is a projection. In case $A$ itself is invertible, we must have $P = Q = A^{- 1}$ .

EXERCISES

Exercise 1. What is the rank of the differentiation operator on $𝒫_{n}$ ? What is its nullity?

Exercise 2. Find the ranks of the following matrices.

$[\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$
$[\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}]$
$[\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}]$
$[\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$

Exercise 3. If $A$ is left multiplication by $P$ on a space of linear transformations (cf. Section: Matrices of transformations , Ex. 5), and if $P$ has rank $m$ , what is the rank of $A$ ?

Exercise 4. The rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks.

Exercise 5.

If $A$ and $B$ are linear transformations on an $n$ -dimensional vector space, and if $A B = 0$ , then $ρ (A) + ρ (B) \leq n$ .
For each linear transformation $A$ on an $n$ -dimensional vector space there exists a linear transformation $B$ such that $A B = 0$ and such that $ρ (A) + ρ (B) = n$ .

Exercise 6. If $A$ , $B$ , and $C$ are linear transformations on a finite-dimensional vector space, then $ρ (A B) + ρ (B C) \leq ρ (B) + ρ (A B C) .$

Exercise 7. Prove that two linear transformations (on the same finite-dimensional vector space) are equivalent if and only if they have the same rank.

Exercise 8.

Suppose that $A$ and $B$ are linear transformations (on the same finite-dimensional vector space) such that $A^{2} = A$ and $B^{2} = B$ . Is it true that $A$ and $B$ are similar if and only if $ρ (A) = ρ (B)$ ?
Suppose that $A$ and $B$ are linear transformations (on the same finite-dimensional vector space) such that $A \neq 0$ , $B \neq 0$ , and $A^{2} = B^{2} = 0$ . Is it true that $A$ and $B$ are similar if and only if $ρ (A) = ρ (B)$ ?

Exercise 9.

If $A$ is a linear transformation of rank one, then there exists a unique scalar $α$ such that $A^{2} = α A$ .
If $α \neq 1$ , then $1 - A$ is invertible.

EXERCISES

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