Question

Prove the converse of Problem I, that is, that if \(M^{\top}=M^{-1}\), then the length of a vector is not changed by the transformation \(M\). Hint : The matrix \(r\) and its transpose \(r^{\mathrm{T}}\) are: $$ r=\left(\begin{array}{lll} x \\ y \\ z \end{array}\right), \quad r^{\tau}=\left(\begin{array}{lll} x & y & z \end{array}\right) $$ Show that \(r^{\mathrm{T}} r=x^{2}+y^{2}+z^{2}\); then use \(r=M r\) and \((1.2)\).

Short answer

The length of a vector is preserved if \(M\) is orthogonal, i.e., \(M^{\top} = M^{-1}\).

Step-by-step solution

- Understand the Problem

The goal is to prove that if a matrix, denoted as \(M\), is orthogonal (i.e., \(M^{\top} = M^{-1}\)), then the length of any vector is preserved under the transformation \(M\). This means that for a vector \( \mathbf{r} \), the length before and after the transformation should be the same.

- Represent the Vector

Given the vector \( \mathbf{r} = \begin{bmatrix} x \ y \ z \end{bmatrix} \) and its transpose \( \mathbf{r}^{\top} = \begin{bmatrix} x & y & z \end{bmatrix}\).

- Calculate the Length of the Vector

To find the length of the vector \( \mathbf{r} \), calculate \( \mathbf{r}^{\top} \mathbf{r} \). \[ \mathbf{r}^{\top} \mathbf{r} = \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = x^2 + y^2 + z^2 \]

- Transform the Vector

When the vector \( \mathbf{r} \) is transformed by the matrix \( M \), the new vector becomes \( \mathbf{r'} = M\mathbf{r} \).

- Calculate the Length of the Transformed Vector

The length of the transformed vector can be found by calculating \( \mathbf{r'}^{\top} \mathbf{r'} \). Since \( \mathbf{r'} = M \mathbf{r} \), \[ \mathbf{r'}^{\top} \mathbf{r'} = (M \mathbf{r})^{\top} (M \mathbf{r}) \]

- Apply the Transpose Property

Using the property of transpose, \[ \mathbf{r'}^{\top} \mathbf{r'} = (M \mathbf{r})^{\top} (M \mathbf{r}) = \mathbf{r}^{\top} M^{\top} M \mathbf{r} \]

- Use Orthogonality

Since \( M \) is orthogonal, by definition \( M^{\top} = M^{-1} \), \[ \mathbf{r'}^{\top} \mathbf{r'} = \mathbf{r}^{\top} M^{-1} M \mathbf{r} = \mathbf{r}^{\top} I \mathbf{r} = \mathbf{r}^{\top} \mathbf{r} \].

- Conclude the Proof

Therefore, the length of \( \mathbf{r'} \) is equal to the length of \( \mathbf{r} \), meaning the transformation by \( M \) does not change the length of the vector. Thus, it is proven that if \( M^{\top} = M^{-1} \), the length of a vector is preserved.

Key concepts

Vector Transformation

A vector transformation involves changing the coordinates of a vector according to certain rules, often given by a matrix. By applying a matrix to a vector, we obtain a new vector that is transformed in a way described by the matrix. In linear algebra, transformations are essential for understanding how vectors change in different contexts. The matrix used for transformation can rotate, scale, or reflect the vector, depending on the properties of the matrix. For example, if we have a matrix \(M\) and a vector \( \textbf{r} \), the transformed vector would be \( \textbf{r'} = M \textbf{r} \).

Vector Length Preservation

Vector length preservation means that after a transformation, the length (or magnitude) of the vector remains the same. This is crucial in many applications, such as physics and computer graphics, where the integrity of the vector's magnitude needs to be maintained. For an orthogonal matrix \(M\) (where \(M^{\top} = M^{-1}\)), it has been proven that the length of the vector does not change. If \( \textbf{r} \) is a vector, its length is calculated as \( \textbf{r}^{\top} \textbf{r} = x^2 + y^2 + z^2 \). After transformation by an orthogonal matrix \(M\), the new vector's length calculation is shown to be the same: \( (\textbf{r'})^{\top} (\textbf{r'}) = \textbf{r}^{\top} \textbf{r} \).

Matrix Transpose

The matrix transpose is an operation that flips a matrix over its diagonal, switching its row and column indices. For a matrix \(M\), its transpose is denoted as \(M^{\top}\). So if \(M\) is a \( 3x3 \) matrix, taking the transpose will swap its rows with its columns.
For example:
\[ M = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}, \quad M^{\top} = \begin{bmatrix} a & d & g \ b & e & h \ c & f & i \end{bmatrix} \]
This property is particularly useful when dealing with transformations and inverses in linear algebra. Specifically, for orthogonal matrices, the transpose is also the inverse (\(M^{\top} = M^{-1}\)).

Matrix Inverse

The inverse of a matrix \(M\), denoted as \(M^{-1}\), is a matrix that, when multiplied by \(M\), results in the identity matrix \(I\). Mathematically, this is expressed as \(MM^{-1} = I\) and \(M^{-1}M = I\). Not all matrices have inverses; only those that are square (same number of rows and columns) and have a non-zero determinant are invertible.
For orthogonal matrices, a special property holds: the transpose of the matrix is equal to its inverse (\(M^{\top} = M^{-1}\)). This makes calculations involving orthogonal matrices simpler and is a crucial concept in proving vector length preservation during transformations.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
Key concepts in linear algebra include:

• Matrices: Rectangular arrays of numbers representing linear transformations.
• Vectors: Elements of vector spaces, often represented as coordinates.
• Vector Transformation: The process of applying a matrix to a vector.
• Orthogonal Matrices: Special matrices that preserve vector lengths (\(M^{\top} = M^{-1}\)).
• Matrix Transpose and Inverse: Operations crucial for solving linear equations and understanding vector transformations.

Understanding these concepts is foundational for fields like computer science, engineering, physics, and more.