Find Eigenvalues and Eigenvectors of Matrix G = A^T A and Orthonormalize Them

Math Problem Statement

Find the eigenvalues and eigenvectors of the matrix G = A superscript T A, and orthonomalise them

Solution

To find the eigenvalues and eigenvectors of the matrix $G = A^T A$ and orthonormalize them, let's follow these steps:

Step 1: Define Matrix $G$

We have already established that:

\begin{bmatrix} ax + bp & ay + bq & az + br \\ cx + dp & cy + dq & cz + dr \\ ex + fp & ey + fq & ez + fr \end{bmatrix}$$ ### Step 2: Find Eigenvalues of $$G$$ The eigenvalues $$\lambda$$ of matrix $$G$$ are found by solving the characteristic equation: $$\text{det}(G - \lambda I) = 0$$ where $$I$$ is the identity matrix of the same dimension as $$G$$. This step involves calculating the determinant of the matrix $$G - \lambda I$$, which results in a cubic equation in $$\lambda$$. ### Step 3: Find Eigenvectors of $$G$$ For each eigenvalue $$\lambda$$, we solve the system of equations: $$(G - \lambda I) \mathbf{v} = 0$$ where $$\mathbf{v}$$ is the eigenvector corresponding to the eigenvalue $$\lambda$$. This involves solving a system of linear equations to find the vectors $$\mathbf{v}$$. ### Step 4: Orthonormalize the Eigenvectors Once we have the eigenvectors, we can orthonormalize them using the **Gram-Schmidt process**. The process ensures that the resulting vectors are orthogonal (perpendicular) and have unit length. 1. **Normalize the first eigenvector $$\mathbf{v}_1$$:** $$\mathbf{u}_1 = \frac{\mathbf{v}_1}{\|\mathbf{v}_1\|}$$ 2. **Orthogonalize the second eigenvector $$\mathbf{v}_2$$ with respect to $$\mathbf{u}_1$$:** $$\mathbf{u}_2 = \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2)$$ where $$\text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) = \frac{\mathbf{v}_2 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1$$ Then normalize $$\mathbf{u}_2$$: $$\mathbf{u}_2 = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|}$$ 3. **Repeat the orthogonalization for the third eigenvector $$\mathbf{v}_3$$ with respect to $$\mathbf{u}_1$$ and $$\mathbf{u}_2$$:** $$\mathbf{u}_3 = \mathbf{v}_3 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3)$$ Normalize $$\mathbf{u}_3$$: $$\mathbf{u}_3 = \frac{\mathbf{u}_3}{\|\mathbf{u}_3\|}$$ ### Summary After following these steps, you'll have the orthonormalized eigenvectors of $$G$$. Would you like a detailed walkthrough with specific numerical examples? ### Related Questions: 1. How do you compute the determinant of a $$3 \times 3$$ matrix? 2. What is the Gram-Schmidt process, and why is it important? 3. How can eigenvalues be used to determine the stability of a system? 4. How do you check if a set of vectors is orthonormal? 5. What is the significance of orthonormal eigenvectors in applications like PCA (Principal Component Analysis)? **Tip:** Always ensure to verify your eigenvectors' orthogonality after normalization to avoid computational errors.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Orthogonalization
Gram-Schmidt Process

Formulas

Eigenvalue equation: det(G - λI) = 0
Gram-Schmidt process for orthonormalization

Theorems

Suitable Grade Level

Advanced Mathematics

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