DLA Discovery guide

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Discovery guide 40.2 Discovery guide

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Recall.

A matrix is self-adjoint if
$\begin{equation*} \inprod{\uvec{u}}{A \uvec{v}} = \inprod{A \uvec{u}}{\uvec{v}} \end{equation*}$
always, so that $\adjoint{A} = A\text{.}$
A matrix is product-preserving (or orthogonal in the real case, unitary in the complex case) if
$\begin{equation*} \inprod{A \uvec{u}}{A \uvec{v}} = \inprod{\uvec{u}}{\uvec{v}} \end{equation*}$
always, so that $\adjoint{A}A = I\text{.}$

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Notation.

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Where necessary, in this discovery guide we will write

$\begin{align*} \amp {\inprod{\blank}{\blank}}_{\R} \text{,} \amp \amp {\inprod{\blank}{\blank}}_{\C} \end{align*}$

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to distinguish between the real and complex dot products, respectively.

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Discovery 40.1. Hermitian eigenvalues.

Suppose $H$ is a Hermitian (i.e. complex self-adjoint) matrix and $\lambda$ and $\uvec{x}$ are an eigenvalue-eigenvector pair for $H\text{,}$ so that

$\begin{equation*} H\uvec{x} = \lambda \uvec{x} \text{.} \end{equation*}$

Use

$\begin{equation*} {\inprod{\uvec{x}}{H \uvec{x}}}_{\C} = {\inprod{H \uvec{x}}{\uvec{x}}}_{\C} \end{equation*}$

to discover something about $\lambda\text{.}$

Hint

Refer to the Algebra rules of complex inner products. “Simplify” each side separately, and then compare the new versions of the two sides again.

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Discovery 40.2. Symmetric eigenvalues.

Convince yourself that a real self-adjoint matrix is also self-adjoint when considered as a complex matrix.

Based on Discovery 40.1, what does this mean about the eigenvalues of a symmetric matrix?

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Discovery 40.3.

The Hermitian matrix

$\begin{equation*} H = \left[\begin{array}{rcc} 0 \amp \ci \amp 0 \\ -\ci \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \end{array}\right] \end{equation*}$

has eigenvalues $\lambda = \pm 1$ with

$\begin{align*} E_{-1}(H) \amp = \Span \left\{ \left[\begin{array}{r} -\ci \\ 1 \\ 0 \end{array}\right] \right\} \text{,} \amp E_1(H) \amp = \Span \left\{ \begin{bmatrix} \ci \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\} \text{.} \end{align*}$

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(a)

Write down a transition matrix $P$ so that $\inv{P} H P$ is diagonal.

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(b)

Write down a unitary transition matrix $U$ so that $\adjoint{U} H U$ is diagonal.

Hint

Statment 4 of Theorem 39.5.6.

Also remember that the complex dot product involves a conjugate.

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(c)

What relationship between the provided eigenvectors of $H$ was crucial to allowing Task b to work?

What if the two provided eigenvectors for $\lambda = 1$ had not initially had that relationship with each other — would you have been able to “fix” it? How?

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Discovery 40.4.

The complex matrix

$\begin{equation*} A = \left[\begin{array}{rrc} 1 \amp 0 \amp 0 \\ -2 \ci \amp -1 \amp 0 \\ -2 \amp 2 \ci \amp 1 \end{array}\right] \end{equation*}$

has eigenvalues $\lambda = \pm 1$ with

$\begin{align*} E_{-1}(A) \amp = \Span \left\{ \begin{bmatrix} 0 \\ \ci \\ 1 \end{bmatrix} \right\} \text{,} \amp E_1(A) \amp = \Span \left\{ \begin{bmatrix} \ci \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\} \text{.} \end{align*}$

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(a)

Write down a transition matrix $P$ so that $\inv{P} A P$ is diagonal.

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(b)

Will you be able to write down a unitary transition matrix $U$ so that $\adjoint{U} A U$ is diagonal? Will trying to “fix” things as in Discovery 40.3.c work?

Hint

The columns of $U$ must be eigenvectors to diagonalize $A\text{!}$

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Let's focus on what seemed to work, and why, in Discovery 40.3.

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Discovery 40.5.

Suppose $H$ is a Hermitian matrix, $\lambda_1,\lambda_2$ are two different eigenvalues for $H\text{,}$ and $\uvec{x}_1,\uvec{x}_2$ are corresponding eigenvectors for those two eigenvalues, respectively.

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(a)

Obtain an expression for $\inprod{H \uvec{x}_1}{\uvec{x}_2}$ in terms of $\lambda_1$ and $\inprod{\uvec{x}_1}{\uvec{x}_2}\text{.}$

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(b)

Use the fact that $H$ is self-adjoint to obtain an expression for $\inprod{H \uvec{x}_1}{\uvec{x}_2}$ in terms of $\lambda_2$ and $\inprod{\uvec{x}_1}{\uvec{x}_2}\text{.}$

Hint

You may wish to make use of the property of eigenvalues of Hermitian matrices found in Discovery 40.1.

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(c)

By comparing your two expressions for $\inprod{H \uvec{x}_1}{\uvec{x}_2}\text{,}$ use the assumption that $\lambda_1 \neq \lambda_2$ to learn something about the eigenvectors $\uvec{x}_1,\uvec{x}_2\text{.}$

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Discovery 40.6.

Determine a unitary matrix $U$ so that $\adjoint{U} A U$ is diagonal, for matrix

$\begin{equation*} A = \begin{bmatrix} 2 \amp 0 \\ 0 \amp \ci \end{bmatrix} \text{.} \end{equation*}$

What is the point of this discovery activity?

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Discovery 40.7.

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(a)

Convince yourself that a diagonal complex matrix $D$ commutes with its adjoint: $\adjoint{D} D = D \adjoint{D}\text{.}$

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(b)

Suppose $A$ is a unitarily diagonalizable complex matrix, so that $\adjoint{U} A U$ is diagonal for some unitary matrix $U\text{.}$

Use the diagonal case from Task a to help verify that $A$ commutes with its adjoint: $\adjoint{A} A = A \adjoint{A}\text{.}$

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(c)

What do you think is the point of this discovery activity?

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Discovery 40.8. Unitary eigenvalues.

Suppose $A$ is a unitary complex matrix and $\lambda$ and $\uvec{x}$ are an eigenvalue-eigenvector pair for $A\text{,}$ so that

$\begin{equation*} A\uvec{x} = \lambda \uvec{x} \text{.} \end{equation*}$

Use

$\begin{equation*} {\inprod{A \uvec{x}}{A \uvec{x}}}_{\C} = {\inprod{\uvec{x}}{\uvec{x}}}_{\C} \end{equation*}$

to discover something about $\lambda\text{.}$

Hint

Refer to the Algebra rules of complex inner products. And remember that a value can only be an eigenvalue if there exist nontrivial corresponding eigenvectors.

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Discovery 40.9. Orthogonal eigenvalues.

Convince yourself that a real orthogonal matrix is also unitary when considered as a complex matrix.

Based on Discovery 40.8, what does this mean about the eigenvalues of an orthogonal matrix?