Math Problems Vault

Problem 5.10.1

If $A$ is a square matrix of index $k>0$ , prove that $\operatorname {index}(A^{k})=1$ .

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Problem 5.10.2

If $A$ is a nilpotent matrix of index $k$ , describe the components in a core-nilpotent decomposition of $A$ .

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Problem 5.10.3

Prove that if $A$ is a symmetric matrix, then $\operatorname {index}(A) \leq 1$ .

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Problem 5.10.4

$A \in \mathbb {C}^{n\times n}$ is said to be a normal matrix whenever $AA^{*}=A^{*}A$ . Prove that if $A$ is normal, then $\operatorname {index}(A) \leq 1$ .

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Remark.

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All symmetric matrices are normal, so this result includes Exercise 5.10.3 as a special case.

Problem 5.10.5

Find a core-nilpotent decomposition and the Drazin inverse of $A = \begin{pmatrix} -2& 0& -4 \\ 4& 2& 4 \\ 3& 2& 2 \end{pmatrix}$ .

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Problem 5.10.6

(a)(b)

For a square matrix $A$ , any scalar $\lambda$ that makes $A-\lambda I$ singular is called an eigenvalue for $A$ . The index of an eigenvalue $\lambda$ is defined to be $\operatorname {index}(\lambda ) = \operatorname {index}(A-\lambda I)$ . Determine the eigenvalues and the index of each eigenvalue for the following matrices:

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(a)

$J = \begin{pmatrix} 1& 0& 0& 0& 0 \\ 0& 1& 0& 0& 0 \\ 0& 0& 1& 0& 0 \\ 0& 0& 0& 2& 0 \\ 0& 0& 0& 0& 2 \end{pmatrix}$ .

(b)

$J = \begin{pmatrix} 1& 1& 0& 0& 0 \\ 0& 1& 1& 0& 0 \\ 0& 0& 1& 0& 0 \\ 0& 0& 0& 2& 1 \\ 0& 0& 0& 0& 2 \end{pmatrix}$ .

Problem 5.10.7

(a)(b)

Let $P$ be a projector different from the identity.

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(a)

Explain why $\operatorname {index}(P)=1$ . What is the index of $I$ ?

(b)

Determine the core-nilpotent decomposition for $P$ .

Problem 5.10.8

Let $N$ be a nilpotent matrix of index $k$ , and suppose that $x$ is a vector such that $N^{k-1}x \neq 0$ . Prove that the set $C = \left\{ x, Nx, N^{2}x, \ldots , N^{k-1}x\right\}$ is a linearly independent set.

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Problem 5.10.9

(a)(b)(c)

Let $A$ be a square matrix of index $k$ , and let $b \in \mathcal{R}(A^{k})$ .

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(a)

Explain why the linear system $Ax=b$ must be consistent.

(b)

Explain why $x=A^{D}b$ is the unique solution in $\mathcal{R}(A^{k})$ .

(c)

Explain why the general solution is given by $A^{D}b + \mathcal{N}(A)$ .

Problem 5.10.10

Suppose that $A$ is a square matrix of index $k$ , and let $A^{D}$ be the Drazin inverse of $A$ . Explain why $AA^{D}$ is the projector onto $\mathcal{R}(A^{k})$ along $\mathcal{N}(A^{k})$ . What does $I-AA^{D}$ project onto and along?

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Problem 5.10.11

(a)(b)(c)

An algebraic group is a set $G$ together with an associative operation between its elements such that $G$ is closed with respect to this operation; $G$ possesses an identity element $E$ (unique); and every member $A \in G$ has an inverse $A^{\# }$ (unique). A matrix group is a set of square matrices that forms an algebraic group under ordinary matrix multiplication.

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(a)

Show that the set of $n\times n$ nonsingular matrices is a matrix group.

(b)

Show that the set of $n\times n$ unitary matrices is a subgroup of the $n\times n$ nonsingular matrices.

(c)

Show that the set $G = \left\{ \begin{pmatrix} \alpha & \alpha \\ \alpha & \alpha \end{pmatrix} \; \middle |\; \alpha \neq 0\right\}$ is a matrix group. In particular, what does the identity element $E \in G$ look like, and what does the inverse $A^{\# }$ of $A \in G$ look like?

Remark.

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These axioms are essentially (A1), (A2), (A4), (A5) in the definition of a vector space.

Problem 5.10.12

For singular matrices, prove that the following statements are equivalent.

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(a)

$A$ is a group matrix (i.e., $A$ belongs to a matrix group).

(b)

$\mathcal{R}(A) \cap \mathcal{N}(A) = 0$ .

(c)

$\mathcal{R}(A)$ and $\mathcal{N}(A)$ are complementary subspaces.

(d)

$\operatorname {index}(A) = 1$ .

(e)

There are nonsingular matrices $Q_{n\times n}$ and $C_{r\times r}$ such that $Q^{-1}AQ = \begin{pmatrix} C_{r\times r}& 0 \\ 0& 0 \end{pmatrix}$ , where $r = \operatorname {rk}\left(A\right)$ .

Problem 5.10.13

(a)(b)

Let $A \in G$ for some matrix group $G$ .

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(a)

Show that the identity element $E \in G$ is the projector onto $\mathcal{R}(A)$ along $\mathcal{N}(A)$ by arguing that $E$ must be of the form $E = Q\begin{pmatrix} I_{r\times r}& 0 \\ 0& 0 \end{pmatrix}Q^{-1}$ .

(b)

Show that the group inverse of $A$ (the inverse of $A$ in $G$ ) must be of the form $A^{\# } = Q\begin{pmatrix} C^{-1}& 0 \\ 0& 0 \end{pmatrix}Q^{-1}$ .