Complementary Subspaces

Let $X$ and $Y$ be subspaces of $\mathbb {R}^{3}$ whose respective bases are $B_{X} = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\right\}$ and $B_{Y} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\right\}$.

Construct an example of a pair of nontrivial complementary subspaces of $\mathbb {R}^{5}$, and explain why your example is valid.

Construct an example to show that if $V = X+Y$ but $X \cap Y \neq 0$, then a vector $v \in V$ can have two different representations as $v = x_{1}+y_{1}$ and $v = x_{2}+y_{2}$, where $x_{1},x_{2} \in X$ and $y_{1},y_{2} \in Y$, but $x_{1} \neq x_{2}$ and $y_{1} \neq y_{2}$.

Explain why $\mathbb {R}^{n \times n} = S \oplus K$, where $S$ and $K$ are the subspaces of $n\times n$ symmetric and skew-symmetric matrices, respectively. What is the projection of $A = \begin{pmatrix} 1& 2& 3 \\ 4& 5& 6 \\ 7& 8& 9 \end{pmatrix}$ onto $S$ along $K$?

For a general vector space, let $X$ and $Y$ be two subspaces with respective bases $B_{X} = \left\{ x_{1},\ldots ,x_{m}\right\}$ and $B_{Y} = \left\{ y_{1},\ldots ,y_{n}\right\}$.

Let $P$ be a projector defined on a vector space $V$. Prove that the range of a projector is the set of its "fixed points," i.e. $\mathcal{R}(P) = \left\{ x \in V \mid Px=x\right\}$.

Suppose that $V = X\oplus Y$, and let $P$ be the projector onto $X$ along $Y$. Prove that $\mathcal{R}(P) = \mathcal{N}(I-P) = X$ and $\mathcal{R}(I-P) = \mathcal{N}(P) = Y$.

Explain why $\left\| P\right\|_{2} \geq 1$ for every projector $P \neq 0$. When is $\left\| P\right\|_{2} = 1$?

Explain why $\left\| I-P\right\|_{2} = \left\| P\right\|_{2}$ for all projectors that are not zero and not equal to the identity.

Prove that if $u,v \in \mathbb {R}^{n}$ are vectors such that $v^{T}u=1$, then $\left\| I-uv^{T}\right\|_{2} = \left\| uv^{T}\right\|_{2} = \left\| u\right\|_{2}\left\| v\right\|_{2} = \left\| uv^{T}\right\|_{F}$.

Suppose that $X$ and $Y$ are complementary subspaces of $\mathbb {R}^{n}$, and let $B = \left[X \mid Y\right]$ be a nonsingular matrix in which the columns of $X$ and $Y$ constitute respective bases for $X$ and $Y$. For an arbitrary vector $v \in \mathbb {R}^{n}$, explain why the projection of $v$ onto $X$ along $Y$ can be obtained by the following two-step process.

1. Solve the system $Bz=v$ for $z$.

2. Partition $z$ as $z = \begin{pmatrix} z_{1} \\ z_{2} \end{pmatrix}$, and set $p = Xz_{1}$.

Let $P$ and $Q$ be projectors.

Prove that $\operatorname {rk}\left(P\right) = \operatorname {trace}(P)$ for every projector $P$ defined on $\mathbb {R}^{n}$.

Let $\left\{ X_{i}\right\}_{i=1}^{k}$ be a collection of subspaces from a vector space $V$, and let $B_{i}$ denote a basis for $X_{i}$. Prove that the following statements are equivalent.

1. $V = X_{1}+X_{2}+\cdots +X_{k}$ and $X_{j}\cap (X_{1}+\cdots +X_{j-1})=0$ for each $j=2,3,\ldots ,k$.

2. For each vector $v \in V$, there is one and only one way to write $v = x_{1}+x_{2}+\cdots +x_{k}$, where $x_{i}\in X_{i}$.

3. $B = B_{1}\cup B_{2}\cup \cdots \cup B_{k}$ with $B_{i}\cap B_{j}=\emptyset$ for $i\neq j$ is a basis for $V$.

For complementary subspaces $X$ and $Y$ of $\mathbb {R}^{n}$, let $P$ be the projector onto $X$ along $Y$, and let $Q = \left[X\mid Y\right]$, where the columns of $X$ and $Y$ constitute bases for $X$ and $Y$, respectively. Prove that if $Q^{-1}A_{n\times n}Q$ is partitioned as $Q^{-1}AQ = \begin{pmatrix} A_{11}& A_{12} \\ A_{21}& A_{22} \end{pmatrix}$, then

Suppose that $\mathbb {R}^{n} = X\oplus Y$, where $\dim X = r$, and let $P$ be the projector onto $X$ along $Y$. Explain why there exist matrices $X_{n\times r}$ and $A_{r\times n}$ such that $P = XA$, where $\operatorname {rk}\left(X\right)=\operatorname {rk}\left(A\right)=r$ and $AX = I_{r}$.

For either a real or complex vector space, let $E$ be the projector onto $X_{1}$ along $Y_{1}$, and let $F$ be the projector onto $X_{2}$ along $Y_{2}$. Prove that $E+F$ is a projector if and only if $EF=FE=0$, and under this condition, prove that $\mathcal{R}(E+F) = X_{1}\oplus X_{2}$ and $\mathcal{N}(E+F) = Y_{1}\cap Y_{2}$.

For either a real or complex vector space, let $E$ be the projector onto $X_{1}$ along $Y_{1}$, and let $F$ be the projector onto $X_{2}$ along $Y_{2}$. Prove that $E-F$ is a projector if and only if $EF=FE=F$, and under this condition, prove that $\mathcal{R}(E-F) = X_{1}\cap Y_{2}$ and $\mathcal{N}(E-F) = Y_{1}\oplus X_{2}$.

For either a real or complex vector space, let $E$ be the projector onto $X_{1}$ along $Y_{1}$, and let $F$ be the projector onto $X_{2}$ along $Y_{2}$. Prove that if $EF=P=FE$, then $P$ is the projector onto $X_{1}\cap X_{2}$ along $Y_{1}+Y_{2}$.

An inner pseudoinverse for $A_{m\times n}$ is a matrix $X_{n\times m}$ such that $AXA=A$, and an outer pseudoinverse for $A$ is a matrix $X$ satisfying $XAX=X$. When $X$ is both an inner and outer pseudoinverse, $X$ is called a reflexive pseudoinverse.