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11.E: Exercises for Chapter 11

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    312
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    Calculational Exercises

    1. Consider \( \mathbb{R}^3 \)with two orthonormal bases: the canonical basis \( e = (e_1 , e_2 , e_3) \) and the basis \(f = (f_1 , f_2 , f_3) \), where
    \[ f1 = \frac{1}{\sqrt{3}}(1,1,1), f_2 = \frac{1}{\sqrt{6}}(1,-2,1), f_3 = \frac{1}{\sqrt{2}}(1,0,-1). \]

    Find the canonical matrix, A, of the linear map \(T \in \cal{L}(\mathbb{R}^3) \) with eigenvectors \(f_1 , f_2 , f_3 \) and eigenvalues 1, 1/2, −1/2, respectively.

    2. For each of the following matrices, verify that \(A\) is Hermitian by showing that \(A = A^*\) ,find a unitary matrix \(U\) such that \(U^{−1}AU\) is a diagonal matrix, and compute \(exp(A)\).
    \[ (a)~ A = \left[ \begin{array}{cc} 4 & 1-i \\ 1+i & 5 \end{array} \right] ~~ (b)~A = \left[ \begin{array}{cc} 3 & -i \\ i & 3 \end{array} \right] ~~ (c)~A = \left[ \begin{array}{cc} 6 & 2+2i \\ 2-2i & 4 \end{array} \right]\]
    \[ (d)~A = \left[ \begin{array}{cc} 0 & 3+i \\ 3-i & -3 \end{array} \right] ~~ (e)~ A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 0 & -1 & -1+i \\ 0 & -1-i & 0 \end{array} \right] ~~ (f)~ A = \left[ \begin{array}{ccc} 2 & \frac{i}{\sqrt{2}} & \frac{-i}{\sqrt{2}} \\ \frac{-i}{\sqrt{2}} & 2 & 0 \\ \frac{i}{\sqrt{2}} & 0 & 2 \end{array} \right] \]

    3. For each of the following matrices, either find a matrix \(P\) (not necessarily unitary) such that \(P^{−1}AP\) is a diagonal matrix, or show why no such matrix exists.

    \[ (a)~ A = \left[ \begin{array}{ccc} 19 & -9 & -6 \\ 25 & -11 & -9 \\ 17 & -9 & -4 \end{array} \right] ~~ (b)~A = \left[ \begin{array}{ccc} -1 & 4 & -2 \\ -3 & 4 & 0 \\ -3 & 1 & 3 \end{array} \right] ~~ (c)~A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 1 & 5 & 0 \\ 0 & 1 & 5 \end{array} \right]\]

    \[ (d)~ A = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 1 \end{array} \right] ~~ (e)~A = \left[ \begin{array}{ccc} -i & 1 & 1 \\ -i & 1 & 1 \\ -i & 1 & 1 \end{array} \right] ~~ (f)~A = \left[ \begin{array}{ccc} 0 & 0 & i \\ 4 & 0 & i \\ 0 & 0 & i \end{array} \right]\]

    4. Let \(r \in \mathbb{R}\) and let \(T \in \cal{L}(\mathbb{C}^2)\) be the linear map with canonical matrix

    \[ T = \left( \begin{array}{cc} 1 & -1 \\ -1 & r \end{array} \right) \]
    (a) Find the eigenvalues of \(T\) .
    (b) Find an orthonormal basis of \( \mathbb{C}^2\) consisting of eigenvectors of \(T\) .
    (c) Find a unitary matrix \(U\) such that \(UT U^*\) is diagonal.

    5. Let \(A\) be the complex matrix given by:
    \[ A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 0 & -1 & -1+i \\ 0 & -1-i & 0 \end{array} \right]\]
    (a) Find the eigenvalues of \(A\).
    (b) Find an orthonormal basis of eigenvectors of \(A\).
    (c) Calculate \(|A| = \sqrt{A^*A}\).
    (d) Calculate \(e^A\) .

    6. Let \(θ \in \mathbb{R}\), and let \(T \in \cal{L}(\mathbb{C}^2) \)have canonical matrix
    \[ M(T ) = \left( \begin{array}{cc} 1 & e^{i\theta} \\ e^{-i\theta} & -1 \end{array} \right). \]
    (a) Find the eigenvalues of \(T\) .
    (b) Find an orthonormal basis for \(\mathbb{C}^2\) that consists of eigenvectors for \(T\) .

    Proof-Writing Exercises

    1. Prove or give a counterexample: The product of any two self-adjoint operators on a finite-dimensional vector space is self-adjoint.

    2. Prove or give a counterexample: Every unitary matrix is invertible.

    3. Let \(V\) be a finite-dimensional vector space over \( \mathbb{F}\), and suppose that \(T \in \cal{L}(V)\) satisfies \(T^2 = T\) . Prove that \(T\) is an orthogonal projection if and only if \(T\) is self-adjoint.

    4. Let \(V\) be a finite-dimensional inner product space over \( \mathbb{C}\), and suppose that \(T \in \cal{L}(V)\) has the property that \(T^* = −T \). (We call T a skew Hermitian operator on \(V\).)
    (a) Prove that the operator \(iT \in \cal{L}(V)\) defined by \((iT )(v) = i(T (v))\), for each \(v \in V\) , is Hermitian.
    (b) Prove that the canonical matrix for \(T\) can be unitarily diagonalized.
    (c) Prove that \(T\) has purely imaginary eigenvalues.

    5. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\), and suppose that \(S, T \in \cal{L}(V)\) are positive operators on \(V\) . Prove that \(S + T\) is also a positive operator on \(T\) .

    6. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\), and let \(T \in \cal{L}(V)\)be any operator on \(V\) . Prove that \(T\) is invertible if and only if 0 is not a singular value of \(T\) .


    This page titled 11.E: Exercises for Chapter 11 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.