Chapter 1: Q6P (page 1)

Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$

Short Answer

Expert verified

$C$ is an orthogonal matrix.

Step by step solution

Given information

C is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors, each of unit length.

Orthogonal matrix

An orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

Calculate the matrix C

Write matrix $C$ as $C = (r_{1}, r_{2})$ , where $r_{1}$ and $r_{2}$ are orthogonal unit vectors, that is,

${r_{1}}^{T} r_{2} = 0 {r_{1}}^{T} r_{1} = 1 {r_{2}}^{T} r_{2} = 1$

Now, calculate $C^{T} C$ .

$C^{T} C = (\begin{matrix} r_{1}^{T} \\ r_{2}^{2} \end{matrix}) (\begin{matrix} r_{1} & r_{2} \end{matrix}) = (\begin{matrix} r_{1}^{T} r_{1} \\ r_{2}^{T} r_{1} \end{matrix} \begin{matrix} r_{1}^{T} r_{2} \\ r_{1}^{T} r_{1} \end{matrix})$

Use the relations above for the orthonormality of $r_{1}$ and $r_{2}$ .

$C^{T} C = (\begin{matrix} 1 & 0 \\ n & 1 \end{matrix})$

The above equation shows that $C$ is an orthogonal matrix.

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Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$

Short Answer

Step by step solution

Given information

Orthogonal matrix

Calculate the matrix C

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Show that if Cis a matrix whose columns are the components (x1,y1)and (x2,y2)of two perpendicular vectors each of unit length, then Cis an orthogonal matrix. Hint: FindCTC

Short Answer

Step by step solution

Given information

Orthogonal matrix

Calculate the matrix C

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

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Study anywhere. Anytime. Across all devices.

Show that if $C$ is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors each of unit length, then $C$ is an orthogonal matrix. Hint: Find $C^{T} C$