The Moore-Penrose Inverse

Show that, for any $m \times n$ matrix $\mathbf{B}$ of full column rank and for any $n \times p$ matrix $\mathbf{C}$ of full row rank,

Show that, for any $m \times n$ matrix $\mathbf{A}$, $\mathbf{A}^{+} = \mathbf{A}'$ if and only if $\mathbf{A}'\mathbf{A}$ is idempotent.

Let $\mathbf{T}$ represent an $m \times p$ matrix, $\mathbf{U}$ an $m \times q$ matrix, $\mathbf{V}$ an $n \times p$ matrix, and $\mathbf{W}$ an $n \times q$ matrix, and define $\mathbf{Q} = \mathbf{W} - \mathbf{V}\mathbf{T}^{-}\mathbf{U}$. If $C(\mathbf{U}) \subset C(\mathbf{T})$ and $R(\mathbf{V}) \subset R(\mathbf{T})$, then

is a generalized inverse of the partitioned matrix $\begin{pmatrix} \mathbf{T} & \mathbf{U} \\ \mathbf{V} & \mathbf{W} \end{pmatrix}$, and

a generalized inverse of $\begin{pmatrix} \mathbf{W} & \mathbf{V} \\ \mathbf{U} & \mathbf{T} \end{pmatrix}$. Show that if the generalized inverses $\mathbf{T}^{-}$ and $\mathbf{Q}^{-}$ (of $\mathbf{T}$ and $\mathbf{Q}$, respectively) are both reflexive [and if $C(\mathbf{U}) \subset C(\mathbf{T})$ and $R(\mathbf{V}) \subset R(\mathbf{T})$], then generalized inverses $(*)$ and $(**)$ are also reflexive.

Determine which of Penrose Conditions (1)--(4) [also known as Moore-Penrose Conditions (1)--(4)] are necessarily satisfied by a left inverse of an $m \times n$ matrix $\mathbf{A}$ (when a left inverse exists). Which of the Penrose conditions are necessarily satisfied by a right inverse of an $m \times n$ matrix $\mathbf{A}$ (when a right inverse exists)?

Let $\mathbf{A}$ represent an $m \times n$ matrix and $\mathbf{G}$ an $n \times m$ matrix.

Show that, for any $m \times n$ matrix $\mathbf{A}$, $(\mathbf{A}^{+}\mathbf{A})^{+} = \mathbf{A}^{+}\mathbf{A}$, and $(\mathbf{A}\mathbf{A}^{+})^{+} = \mathbf{A}\mathbf{A}^{+}$.

Show that, for any $n \times n$ symmetric matrix $\mathbf{A}$, $\mathbf{A}\mathbf{A}^{+} = \mathbf{A}^{+}\mathbf{A}$.

Let $\mathbf{V}$ represent an $n \times n$ symmetric nonnegative definite matrix, $\mathbf{X}$ an $n \times p$ matrix, and $\mathbf{d}$ a $p \times 1$ vector. Using the results of Exercises 8 and 19.11 (or otherwise), show that, for the vector $\mathbf{X}(\mathbf{X}'\mathbf{X})^{-}\mathbf{d}$ to be a solution, for every $\mathbf{d} \in C(\mathbf{X}')$, to the problem of minimizing the quadratic form $\mathbf{a}'\mathbf{V}\mathbf{a}$ (in $\mathbf{a}$) subject to $\mathbf{X}'\mathbf{a} = \mathbf{d}$, it is necessary and sufficient that $C(\mathbf{V}^{+}\mathbf{X}) \subset C(\mathbf{X})$.

Let $\mathbf{A}$ represent an $n \times n$ matrix. Show that if $\mathbf{A}$ is symmetric and positive semidefinite, then $\mathbf{A}^{+}$ is symmetric and positive semidefinite and that if $\mathbf{A}$ is symmetric and positive definite, then $\mathbf{A}^{+}$ is symmetric and positive definite. Do so by taking advantage of the result that if $\mathbf{A}$ is symmetric and nonnegative definite (and nonnull), then $\mathbf{A}^{+} = \mathbf{T}^{+}(\mathbf{T}^{+})'$ for any matrix $\mathbf{T}$ of full row rank (and with $n$ columns) such that $\mathbf{A} = \mathbf{T}'\mathbf{T}$.

Let $\mathbf{C}$ represent an $m \times n$ matrix. Show that, for any $m \times m$ idempotent matrix $\mathbf{A}$, $(\mathbf{A}\mathbf{C})^{+}\mathbf{A}' = (\mathbf{A}\mathbf{C})^{+}$ and that, for any $n \times n$ idempotent matrix $\mathbf{B}$, $\mathbf{B}'(\mathbf{C}\mathbf{B})^{+} = (\mathbf{C}\mathbf{B})^{+}$.

Let $\mathbf{a}$ represent an $n \times 1$ vector of variables, and impose on $\mathbf{a}$ the constraint $\mathbf{X}'\mathbf{a} = \mathbf{d}$, where $\mathbf{X}$ is an $n \times p$ matrix and $\mathbf{d}$ a $p \times 1$ vector such that $\mathbf{d} \in C(\mathbf{X}')$. And, define $f(\mathbf{a}) = \mathbf{a}'\mathbf{V}\mathbf{a} - 2\mathbf{b}'\mathbf{a}$, where $\mathbf{V}$ is an $n \times n$ symmetric nonnegative definite matrix and $\mathbf{b}$ is an $n \times 1$ vector such that $\mathbf{b} \in C(\mathbf{V}, \mathbf{X})$. Further, let $\mathbf{R}$ represent any matrix such that $\mathbf{V} = \mathbf{R}'\mathbf{R}$, let $\mathbf{a}_{0}$ represent any $n \times 1$ vector such that $\mathbf{X}'\mathbf{a}_{0} = \mathbf{d}$, and take $\mathbf{s}$ to be any $n \times 1$ vector such that $\mathbf{b} = \mathbf{V}\mathbf{s} + \mathbf{X}\mathbf{t}$ for some $p \times 1$ vector $\mathbf{t}$. Show that $f(\mathbf{a})$ attains its minimum value (under the constraint $\mathbf{X}'\mathbf{a} = \mathbf{d}$) at a point $\mathbf{a}_{*}$ if and only if

for some $n \times 1$ vector $\mathbf{w}$. Do so by, for instance, using the results of Exercise 11 in combination with the result that, for any $n \times k$ matrix $\mathbf{Z}$ whose columns span $N(\mathbf{X}')$, $f(\mathbf{a})$ attains its minimum value (subject to the constraint $\mathbf{X}'\mathbf{a} = \mathbf{d}$) at a point $\mathbf{a}_{*}$ if and only if

for some $k \times 1$ vector $\mathbf{w}$.

Let $\mathbf{A}$ represent an $n \times n$ symmetric nonnegative definite matrix, and let $\mathbf{B}$ represent an $n \times n$ matrix. Suppose that $\mathbf{B} - \mathbf{A}$ is symmetric and nonnegative definite (in which case $\mathbf{B}$ is symmetric and nonnegative definite). Show that $\mathbf{A}^{+} - \mathbf{B}^{+}$ is nonnegative definite if and only if $\operatorname {rk}\left(\mathbf{A}\right) = \operatorname {rk}\left(\mathbf{B}\right)$. Do so by, for instance, using the results of Exercises 1 and 18.15, the result that $\mathbf{W}^{-1} - \mathbf{V}^{-1}$ is nonnegative definite for any $m \times m$ symmetric positive definite matrices $\mathbf{W}$ and $\mathbf{V}$ such that $\mathbf{V} - \mathbf{W}$ is nonnegative definite, and the result that the Moore-Penrose inverse $\mathbf{H}^{+}$ of a $k \times k$ symmetric nonnegative definite matrix $\mathbf{H}$ equals $\mathbf{T}^{+}(\mathbf{T}^{+})'$, where $\mathbf{T}$ is any matrix of full row rank (and with $k$ columns) such that $\mathbf{H} = \mathbf{T}'\mathbf{T}$.