Question

Normalize the rows to make each of the following matrices orthogonal. a. \(A=\left[\begin{array}{rl}1 & 1 \\ -1 & 1\end{array}\right]\) b. \(A=\left[\begin{array}{rr}3 & -4 \\ 4 & 3\end{array}\right]\) c. \(A=\left[\begin{array}{rr}1 & 2 \\ -4 & 2\end{array}\right]\) d. \(A=\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right],(a, b) \neq(0,0)\) e. \(A=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 2\end{array}\right]\) f. \(A=\left[\begin{array}{rrr}2 & 1 & -1 \\ 1 & -1 & 1 \\ 0 & 1 & 1\end{array}\right]\) g. \(A=\left[\begin{array}{rrr}-1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1\end{array}\right]\) h. \(A=\left[\begin{array}{rrr}2 & 6 & -3 \\ 3 & 2 & 6 \\ -6 & 3 & 2\end{array}\right]\)

Short answer

Normalize each row of the matrix by dividing it by its Euclidean norm.

Step-by-step solution

Finding Row Norms for Matrix a

For matrix \(A = \begin{bmatrix} 1 & 1 \ -1 & 1 \end{bmatrix}\), calculate the norm of each row. The norm of the first row is:\[ \| (1, 1) \| = \sqrt{1^2 + 1^2} = \sqrt{2} \]The norm of the second row is:\[ \| (-1, 1) \| = \sqrt{(-1)^2 + 1^2} = \sqrt{2} \]

Normalizing Rows for Matrix a

Divide each element of a row by its norm to normalize:Normalized first row:\[ \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]Normalized second row:\[ \left( \frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]Thus, the orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \ \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \]

Row Norms for Matrix b

For matrix \(A = \begin{bmatrix} 3 & -4 \ 4 & 3 \end{bmatrix}\), calculate the row norms:The norm of the first row:\[ \| (3, -4) \| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5 \]The norm of the second row:\[ \| (4, 3) \| = \sqrt{4^2 + 3^2} = \sqrt{25} = 5 \]

Normalizing Rows for Matrix b

Divide each element by the row's norm:Normalized first row:\[ \left( \frac{3}{5}, \frac{-4}{5} \right) \]Normalized second row:\[ \left( \frac{4}{5}, \frac{3}{5} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{3}{5} & \frac{-4}{5} \ \frac{4}{5} & \frac{3}{5} \end{bmatrix} \]

Row Norms for Matrix c

Calculate the norms for matrix \(A = \begin{bmatrix} 1 & 2 \ -4 & 2 \end{bmatrix}\):First row:\[ \| (1, 2) \| = \sqrt{1^2 + 2^2} = \sqrt{5} \]Second row:\[ \| (-4, 2) \| = \sqrt{(-4)^2 + 2^2} = \sqrt{20} = 2\sqrt{5} \]

Normalizing Rows for Matrix c

Normalize each row of matrix \(A\):Normalized first row:\[ \left( \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) \]Normalized second row:\[ \left( \frac{-4}{2\sqrt{5}}, \frac{2}{2\sqrt{5}} \right) = \left( \frac{-2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} \]

Row Norms for Matrix d

For matrix \(A = \begin{bmatrix} a & b \ -b & a \end{bmatrix}\), compute the norms:First row:\[ \| (a, b) \| = \sqrt{a^2 + b^2} \]Second row:\[ \| (-b, a) \| = \sqrt{b^2 + a^2} = \sqrt{a^2 + b^2} \]

Normalizing Rows for Matrix d

Normalize each row:First row:\[ \left( \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right) \]Second row:\[ \left( \frac{-b}{\sqrt{a^2 + b^2}}, \frac{a}{\sqrt{a^2 + b^2}} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{a}{\sqrt{a^2 + b^2}} & \frac{b}{\sqrt{a^2 + b^2}} \ \frac{-b}{\sqrt{a^2 + b^2}} & \frac{a}{\sqrt{a^2 + b^2}} \end{bmatrix} \]

Normalizing Last Row for Matrix e

Matrix \(A = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 2 \end{bmatrix}\) already has orthogonal first two rows due to being rotations. Normalize the third row:\[ \| (0, 0, 2) \| = 2 \]Normalized third row:\[ (0, 0, 1) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{bmatrix} \]

Row Norms for Matrix f

For matrix \(A = \begin{bmatrix} 2 & 1 & -1 \ 1 & -1 & 1 \ 0 & 1 & 1 \end{bmatrix}\), calculate the norms:First row:\[ \| (2, 1, -1) \| = \sqrt{2^2 + 1^2 + (-1)^2} = \sqrt{6} \]Second row:\[ \| (1, -1, 1) \| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \]Third row:\[ \| (0, 1, 1) \| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

Normalizing Rows for Matrix f

Normalize each row:First row:\[ \left( \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-1}{\sqrt{6}} \right) \]Second row:\[ \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \]Third row:\[ \left( 0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{2}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{-1}{\sqrt{6}} \ \frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \]

Row Norms for Matrix g

For matrix \(A = \begin{bmatrix} -1 & 2 & 2 \ 2 & -1 & 2 \ 2 & 2 & -1 \end{bmatrix}\), calculate the norms:First row:\[ \| (-1, 2, 2) \| = \sqrt{(-1)^2 + 2^2 + 2^2} = \sqrt{9} = 3 \]Second row:\[ \| (2, -1, 2) \| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{9} = 3 \]Third row:\[ \| (2, 2, -1) \| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{9} = 3 \]

Normalizing Rows for Matrix g

Normalize each row:First row:\[ \left( \frac{-1}{3}, \frac{2}{3}, \frac{2}{3} \right) \]Second row:\[ \left( \frac{2}{3}, \frac{-1}{3}, \frac{2}{3} \right) \]Third row:\[ \left( \frac{2}{3}, \frac{2}{3}, \frac{-1}{3} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{-1}{3} & \frac{2}{3} & \frac{2}{3} \ \frac{2}{3} & \frac{-1}{3} & \frac{2}{3} \ \frac{2}{3} & \frac{2}{3} & \frac{-1}{3} \end{bmatrix} \]

Row Norms for Matrix h

For matrix \(A = \begin{bmatrix} 2 & 6 & -3 \ 3 & 2 & 6 \ -6 & 3 & 2 \end{bmatrix}\), calculate the norms:First row:\[ \| (2, 6, -3) \| = \sqrt{2^2 + 6^2 + (-3)^2} = \sqrt{49} = 7 \]Second row:\[ \| (3, 2, 6) \| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{49} = 7 \]Third row:\[ \| (-6, 3, 2) \| = \sqrt{(-6)^2 + 3^2 + 2^2} = \sqrt{49} = 7 \]

Normalizing Rows for Matrix h

Normalize each row:First row:\[ \left( \frac{2}{7}, \frac{6}{7}, \frac{-3}{7} \right) \]Second row:\[ \left( \frac{3}{7}, \frac{2}{7}, \frac{6}{7} \right) \]Third row:\[ \left( \frac{-6}{7}, \frac{3}{7}, \frac{2}{7} \right) \]The orthogonal matrix is:\[ A_\text{orth} = \begin{bmatrix} \frac{2}{7} & \frac{6}{7} & \frac{-3}{7} \ \frac{3}{7} & \frac{2}{7} & \frac{6}{7} \ \frac{-6}{7} & \frac{3}{7} & \frac{2}{7} \end{bmatrix} \]

Key concepts

Matrix Normalization

Matrix normalization is a key step needed to transform any given matrix into an orthogonal matrix. An orthogonal matrix has rows and columns that are unit vectors, meaning each vector's magnitude is equal to one. To achieve this, the process involves normalizing each row within a matrix. This entails dividing each element of a row by the row’s norm, ensuring that the length of each resulting vector is one.

The norm of a vector (or row in this context) is typically calculated using the Euclidean norm, represented as \( \| \mathbf{x} \| = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} \). By dividing each element of a row by its corresponding row norm, we effectively scale the vector to have a unit length. This is called normalizing the row.

Normalization doesn’t change the direction of the vector; it only adjusts its length to one. Matrices with normalized rows are easier to work with in many mathematical and computational applications, including solving systems of linear equations and performing various transformations in 3D graphics.

Row Norms

The concept of row norms is fundamental to understanding how normalization works. The row norm of a given vector from a matrix is calculated using the Euclidean norm, which is the square root of the sum of the squares of its components.

For example, the norm \( \| (a, b) \| \) is computed as \( \sqrt{a^2 + b^2} \). Each row of a matrix is treated independently, and its norm is calculated to determine the scaling factor for normalization.

• Calculating norms is critical when converting matrices into orthogonal matrices.
• The row norm helps in right-sizing the row during normalization.

Understanding and calculating row norms is essential in many fields of mathematics and machine learning, where metrics like distance and similarity are crucial.

Orthogonalization Process

The orthogonalization process converts a given matrix into one with orthogonal rows and columns. An orthogonal matrix has distinct advantages, including simplified matrix operations like inversion and multiplication.

• Having orthogonal rows implies that each row vector is perpendicular to others, which is a highly desirable property in computations.
• Orthogonal matrices enhance numerical stability for various algorithms, making them preferable in computer simulations.

Gaining an orthogonal matrix usually involves first normalizing rows (as discussed) and sometimes applying further transformations, like the Gram-Schmidt process. This is especially true when the original matrix is not already in a form nearing orthogonality.

The orthogonalization process helps programmatically in creating rotation matrices in graphics, ensuring no distortion occurs.

Linear Algebra in Computer Graphics

Linear algebra plays a pivotal role in computer graphics, enabling the creation, transformation, and manipulation of graphical objects on the screen. One core component of this application is the use of matrices, particularly orthogonal matrices, to perform complex transformations.

In computer graphics:

• Transformation matrices are used to perform operations such as translation, scaling, and rotation of objects.
• An orthogonal matrix is especially vital in 3D graphics for operations like rotating objects without altering their shape.

Orthogonal matrices represent rotations perfectly, as they preserve angles and distances. This means when you rotate an object using an orthogonal matrix, its size and form remain unchanged, a critical requirement for realistic rendering in graphics environments.

Additionally, orthogonal transformations are computationally efficient, an advantage for rendering pipelines that need to process vast amounts of data quickly to render real-time 3D animations.