Chapter 3: Q10P (page 159)

Show that $Tr (C^{- 1} MC) = Tr M$ . Hint: See $(9.13)$ . Thus, show that the sum of the eigenvalues of $M$ is equal to $Tr M$ .

Short Answer

Expert verified

The total of a matrix's eigen values is the matrix's trace.

Step by step solution

Given information from question

The matrices $M = (\begin{matrix} 5 & - 2 \\ - 2 & 2 \end{matrix})$ and $C = (\begin{matrix} \frac{1}{\sqrt{5}} & \frac{- 2}{\sqrt{5}} \\ \frac{- 2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix})$

Eigen Values

Eigenvalues are a set of scalar values that are related with a set of linear equations, and are most typically seen in matrix equations. Eigenvectors are also known as characteristic roots. Once linear transformations are applied, it's a non-zero vector that can only be changed by its scalar factor.

Calculate the trace of a matrix is equal to the sum of its eigen values

$C^{- 1} = (\begin{matrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix}) = T r ((\begin{matrix} \frac{- 1}{\sqrt{5}} & \frac{- 2}{\sqrt{5}} \\ \frac{- 2}{\sqrt{5}} & \frac{- 1}{\sqrt{5}} \end{matrix}) \times (\begin{matrix} \frac{1}{\sqrt{5}} & \frac{- 2}{\sqrt{5}} \\ \frac{- 2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{matrix}) \times (\begin{matrix} 5 & - 2 \\ - 2 & 2 \end{matrix})) = T r ((\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) \times (\begin{matrix} 5 & - 2 \\ - 2 & 2 \end{matrix})) = T r ((\begin{matrix} 5 & - 2 \\ - 2 & 2 \end{matrix}))$

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

But since we know D is a diagonal matrix, and its diagonal elements are the eigenvalues for the matrix $M$ , therefore the trace of a matrix is equal to the sum of its eigen values.

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Show that $Tr (C^{- 1} MC) = Tr M$ . Hint: See $(9.13)$ . Thus, show that the sum of the eigenvalues of $M$ is equal to $Tr M$ .

Short Answer

Step by step solution

Given information from question

Eigen Values

Calculate the trace of a matrix is equal to the sum of its eigen values

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Show that Tr(C-1MC)=TrM. Hint: See (9.13). Thus, show that the sum of the eigenvalues of M is equal to TrM.

Short Answer

Step by step solution

Given information from question

Eigen Values

Calculate the trace of a matrix is equal to the sum of its eigen values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Magnetism and Electromagnetic Induction

Fields in Physics

Astrophysics

Classical Mechanics

Famous Physicists

Study anywhere. Anytime. Across all devices.

Show that $Tr (C^{- 1} MC) = Tr M$ . Hint: See $(9.13)$ . Thus, show that the sum of the eigenvalues of $M$ is equal to $Tr M$ .