Section 10.2

Supplementary Problems

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Problem 10.2.1

Consider the nonnormal invertible 2×22 \times 2 matrix

A=(110−1). A = \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}.

Find A2A^{2}, A3A^{3} and A−1A^{-1} utilizing the Cayley-Hamilton theorem.

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Problem 10.2.2

Let ϵ∈R\epsilon \in \mathbb {R} and ϵ≠0\epsilon \neq 0. Let

A(ϵ)=(0ϵϵ0). A(\epsilon ) = \begin{pmatrix} 0 & \epsilon \\ \epsilon & 0 \end{pmatrix}.
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(i)

Calculate cos⁡(A(ϵ))\cos (A(\epsilon )) and cos⁡(A(ϵ)⊗A(ϵ))\cos (A(\epsilon ) \otimes A(\epsilon )) applying the Cayley-Hamilton theorem.

(ii)

Calculate tanh⁡(A(ϵ))\tanh (A(\epsilon )) applying the Cayley-Hamilton theorem.

Problem 10.2.3

Let ϵ∈R\epsilon \in \mathbb {R}. Consider the 4×44 \times 4 matrix

A(ϵ)=(0ϵϵϵ)⊗(0ϵϵϵ). A(\epsilon ) = \begin{pmatrix} 0 & \epsilon \\ \epsilon & \epsilon \end{pmatrix} \otimes \begin{pmatrix} 0 & \epsilon \\ \epsilon & \epsilon \end{pmatrix}.

Find exp⁡(A(ϵ))\exp (A(\epsilon )) applying the Cayley-Hamilton theorem.

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Problem 10.2.4

Consider the Bell matrix

B=12(1001011001−10100−1) B = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{pmatrix}

which is a unitary matrix. Apply the Cayley-Hamilton theorem to find the skew-hermitian matrix KK such that B=eKB = e^{K}.

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