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Linear Algebra in Action

Harry Dym

Chapter 11

Pseudoinverses - all with Video Answers

Educators

Chapter Questions

03:31

Problem 1

Let $A$ be a $4 \times 5$ matrix such that $E P A=U$ is an upper echelon matrix with pivots in the 11,22 and 34 positions. Show that there exists an invertible $5 \times 5$ lower triangular matrix $F$ and a $5 \times 5$ permutation matrix $\Pi$ such that
$$
\Pi F U^T=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right] \text { and } A=(E P)^{-1}\left[\begin{array}{cc}
I_3 & O_{3 \times 2} \\
O_{1 \times 3} & O_{1 \times 2}
\end{array}\right]\left(F^T \Pi^T\right)^{-1} \text {. }
$$

Victor Salazar
Victor Salazar
Numerade Educator

Problem 2

Verify assertions (1)-(3) of Lemma 11.2 via the decompositions (11.4) and (11.5).

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Problem 3

Verify the formula for $\mathcal{N}_A$ that is given in Lemma 11.5.

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Problem 4

Use formulas (11.7)-(11.10) to confirm that
$$
\mathcal{R}_A \cap \mathcal{N}_{A^{\circ}}=\{0\} \text { and } \mathcal{R}_{A^{\circ}} \cap \mathcal{N}_A=\{0\} \text {. }
$$

Victor Salazar
Victor Salazar
Numerade Educator
02:00

Problem 5

Show that, in the setting of Lemma 11.5,
$$
\mathcal{R}_A \text { is orthogonal to } \mathcal{N}_{A^0} \Longleftrightarrow B_1=O_{r \times(p-r)}
$$
and
$$
\mathcal{N}_A \text { is orthogonal to } \mathcal{R}_{A^{\circ}} \Longleftrightarrow B_2=O_{(q-r) \times r} \text {. }
$$

Vikash Ranjan
Vikash Ranjan
Numerade Educator

Problem 6

Let $A \in \mathbb{F}^{p \times q}$ and suppose that $\mathcal{R}_A=\mathbb{F}^p$ and $\mathcal{N}_A \dot{+} \mathcal{Y}=$ $\mathbb{F}^q$ for some proper nonzero subspace $\mathcal{Y}$ of $\mathbb{F}^q$. Show that there exists a pseudoinverse $A^{\circ}$ of $A$ such that $\mathcal{N}_{A^{\circ}}=\{\mathbf{0}\}$ and $\mathcal{R}_{A^{\circ}}=\mathcal{Y}$.

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Problem 7

Let
$$
A=\left[\begin{array}{lll}
1 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right], \mathcal{X}=\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]\right\} \text { and } \mathcal{Y}=\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
0 \\
1 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
1 \\
1 \\
0
\end{array}\right]\right\}
$$
Find a pseudoinverse $A^{\circ}$ of the matrix $A$ such that $\mathcal{N}_{A^{\circ}}=\mathcal{Y}$ and $\mathcal{R}_{A^{\circ}}=\mathcal{X}$.

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02:23

Problem 8

Let $A \in \mathbb{C}^{p \times q}$ admit a singular value decomposition of the form $A=V S U^H$, where $V \in \mathbb{C}^{p \times p}$ and $U \in \mathbb{C}^{q \times q}$ are both unitary. Suppose further that rank $A=r, S=\operatorname{diag}\left\{D, O_{(p-r) \times(q-r)}\right\}$ and that $1 \leq$ $r<\min \{p, q\}$.
(1) Find formulas for $A^H A, A A^H, A A^{\dagger}$ and $A^{\dagger} A$.
(2) Show that the ranges of $A^H A$ and $A^{\dagger} A$ coincide.
(3) Show that the ranges of $A A^H$ and $A A^{\dagger}$ coincide.
(4) Describe the null spaces of the four matrices considered in (1) in terms of appropriately chosen sub-blocks of $U$ and $V$.

Jack Chen
Jack Chen
Numerade Educator

Problem 9

Show that if $A \in \mathbb{C}^{p \times p}, B \in \mathbb{C}^{p \times q}$ and $\mathcal{R}_B \subseteq \mathcal{R}_A$, then $A A^{\dagger} B=B$.

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Problem 10

Show that $\mathcal{R}_{A^{\dagger}}=\mathcal{R}_{A^H}$ and $\mathcal{N}_{A^{\dagger}}=\mathcal{N}_{A^H}$.

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Problem 11

Show that if $A \in \mathbb{C}^{p \times q}$, then the matrix $A A^{\dagger}$ is an orthogonal projection from $\mathbb{C}^p$ onto $\mathcal{R}_A$.

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02:49

Problem 12

Use the representation formulas (11.14) and (11.16) to give a new proof of the following two formulas, for any matrix $A \in \mathbb{C}^{p \times q}$ :
(1) $\mathbb{C}^p=\mathcal{R}_A \oplus \mathcal{N}_{A^H}$ (with respect to the standard inner product).
(2) $\mathbb{C}^q=\mathcal{R}_{A^H} \oplus \mathcal{N}_A$ (with respect to the standard inner product).

Victor Salazar
Victor Salazar
Numerade Educator

Problem 13

Show that if $B, C \in \mathbb{C}^{p \times q}, A \in \mathbb{C}^{q \times q}$ and rank $B=$ $\operatorname{rank} C=\operatorname{rank} A=q$, then
$$
\left(B A C^H\right)^{\dagger}=C\left(C^H C\right)^{-1} A^{-1}\left(B^H B\right)^{-1} B^H
$$
and give explicit formulas for $\left(B A C^H\right)^{\dagger}\left(B A C^H\right)$ and $\left(B A C^H\right)\left(B A C^H\right)^{\dagger}$ in terms of $B, B^H, C$ and $C^H$.

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02:13

Problem 14

Show that if $A \in \mathbb{C}^{p \times q}$, then $A^{\dagger} A A^H=A^H A A^{\dagger}=A^H$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator

Problem 15

Show that if
$$
E=\left[\begin{array}{cc}
O & B \\
B^H & C
\end{array}\right]
$$
is a Hermitian matrix such that $\mathcal{R}_C \subseteq \mathcal{R}_{B^H}$, then the Moore-Penrose inverse $E^{\dagger}$ of $E$ is given by the formula
$$
E^{\dagger}=\left[\begin{array}{cc}
-\left(B^{\dagger}\right)^H C B^{\dagger} & \left(B^{\dagger}\right)^H \\
B^{\dagger} & O
\end{array}\right]
$$

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03:02

Problem 16

Let
$$
C=B\left[\begin{array}{ll}
A & O \\
O & O
\end{array}\right] B^H,
$$
where $B$ is invertible, and let $A^{\dagger}$ denote the Moore-Penrose inverse of $A$. Show that the matrix
$$
\left(B^{-1}\right)^H\left[\begin{array}{cc}
A^{\dagger} & O \\
O & O
\end{array}\right] B^{-1}
$$
is a pseudoinverse of $C$, but it is not a Moore-Penrose inverse.

Andrija Isakov
Andrija Isakov
Numerade Educator
17:50

Problem 17

Show that the matrix
$$
\left[\begin{array}{cc}
A A^{\dagger} & O \\
B^H A^{\dagger} & O
\end{array}\right]
$$
is a projection but not an orthogonal projection with respect to the standard inner product (unless $B^H A^{\dagger}=O$ ).

Chris Trentman
Chris Trentman
Numerade Educator
03:07

Problem 18

Let $A_1, A_2 \in \mathbb{C}^{p \times q}$ and $B_1, B_2 \in \mathbb{C}^{p \times r}$ and suppose that $\mathcal{R}_{B_1} \subseteq \mathcal{R}_{A_1}$ and $\mathcal{R}_{B_2} \subseteq \mathcal{R}_{A_2}$. Show by example that this does not imply that $\mathcal{R}_{B_1+B_2} \subseteq \mathcal{R}_{A_1+A_2}$. [HINT: Try $B_i=\mathbf{u}_i \mathbf{v}_i^H$ and $A_i=\mathbf{u}_i \mathbf{w}_i^H$ for $i=1,2$, with $\mathbf{v}_1$ orthogonal to $\mathbf{v}_2$ and $\mathbf{w}_1=\mathbf{w}_2$.]

Mahnoor Amin
Mahnoor Amin
Numerade Educator

Problem 19

Let $A \in \mathbb{C}^{p \times p}, B \in \mathbb{C}^{p \times q}$,
$$
M=\left[\begin{array}{cc}
A & B \\
B^H & O
\end{array}\right]
$$
and suppose that $B B^H=I_p$. Show that the Moore-Penrose inverse
$$
M^{\dagger}=\left[\begin{array}{cc}
O & B \\
B^H & -B^H A B
\end{array}\right] .
$$

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17:50

Problem 20

Let $B \in \mathbb{C}^{p \times q}$. Show that:
(1) $B^{\dagger} B$ is the orthogonal projection of $\mathbb{C}^q$ onto $\mathcal{R}_{B^H}$.
(2) $B B^{\dagger}$ is the orthogonal projection of $\mathbb{C}^p$ onto $\mathcal{R}_B$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 21

Let $A \in \mathbb{F}^{p \times q}$. Show that if $\operatorname{rank} A=q$, then $A^H A$ is invertible and
$$
\left(A^H A\right)^{-1} A^H=A^{\dagger} .
$$

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