Chapter 3: Q34P (page 160)

By multiplying out $M = C D C^{- 1}$ where C is the rotation matrix (11.14) and D is the diagonal matrix
$(\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix})$
Show that if M can be diagonalized by a rotation, then M is symmetric.

Short Answer

Expert verified

$M$ can be diagonalized by a rotation and $M$ is symmetric.

Step by step solution

Given Information

The matrices C and D are given.

$C = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) a n d D = (\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix})$

Definition of Rotational Matrix

A rotation matrix is a transformation matrix that acts on a vector and outputs a rotated vector while keeping the coordinate axes constant.

Rotational Matrix and check for symmetry

The formula for rotational matrix is mathematically presented as,

$M = C D C^{- 1} M^{T} = {(C D C^{- 1})}^{T} = {(C^{T})}^{- 1} D^{T} C^{T}$

Here, C is a rotational matrix, which means $C^{T} = C^{- 1}$ and D is the diagonal, which means $D^{T} = D$ .

$D^{T} = {(\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix})}^{T} = (\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix}) M^{T} = {(C^{- 1})}^{- 1} D C^{- 1} \Rightarrow C D C^{- 1} \Rightarrow M^{T} = M$

This proves that M is symmetric.

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By multiplying out $M = C D C^{- 1}$ where C is the rotation matrix (11.14) and D is the diagonal matrix
$(\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix})$
Show that if M can be diagonalized by a rotation, then M is symmetric.

Short Answer

Step by step solution

Given Information

Definition of Rotational Matrix

Rotational Matrix and check for symmetry

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By multiplying out M=CDC-1 where C is the rotation matrix (11.14) and D is the diagonal matrix(x100x2)Show that if M can be diagonalized by a rotation, then M is symmetric.

Short Answer

Step by step solution

Given Information

Definition of Rotational Matrix

Rotational Matrix and check for symmetry

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Force

Astrophysics

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Linear Momentum

Quantum Physics

Study anywhere. Anytime. Across all devices.

By multiplying out $M = C D C^{- 1}$ where C is the rotation matrix (11.14) and D is the diagonal matrix
$(\begin{matrix} x_{1} & 0 \\ 0 & x_{2} \end{matrix})$
Show that if M can be diagonalized by a rotation, then M is symmetric.