91Ó°ÊÓ

To find the eigenvalues of a matrix, we use the characteristic polynomial. This polynomial is derived from the determinant of \(A - \lambda I\), where \A\ is your matrix and \lambda\ is a scalar (an eigenvalue). For our matrix A:\[A = \left[\begin{array}{rrr}4 & 1 & -2 \1 & 4 & -2 \-2 & -2 & 7\end{array}\right]\]we form \(A - \lambda I\) by subtracting \lambda\ from each of the diagonal entries of A. This gives us a new matrix:\[A - \lambda I = \left[\begin{array}{rrr} 4-\lambda & 1 & -2 \ 1 & 4-\lambda & -2 \ -2 & -2 & 7-\lambda \end{array}\right]\]Next, we compute the determinant of this matrix to obtain the characteristic polynomial. Simplifying the determinant:\[\begin{vmatrix} 4-\lambda & 1 & -2 \ 1 & 4-\lambda & -2 \ -2 & -2 & 7-\lambda \end{vmatrix}\]By expanding it (which we'll cover next), we find the characteristic polynomial.

The determinant of a matrix can be computed using cofactor expansion. This method breaks down a large determinant into smaller, more manageable parts. For our matrix \(A - \lambda I\), we expand along the first row:\[(4-\lambda) \cdot \begin{vmatrix} 4 - \lambda & -2 \ -2 & 7 - \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -2 \ -2 & 7 - \lambda \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 4 - \lambda \ -2 & -2 \end{vmatrix}\]We then find the determinant of each 2x2 matrix, and simplify each term. After computing and combining these determinants, our characteristic polynomial simplifies to:\[\lambda^3 - 15\lambda^2 + 63\lambda - 67 = 0\]Solving this cubic polynomial, we obtain the eigenvalues: \lambda = 7, 3, 5.

After finding the eigenvectors associated with each eigenvalue, we often need to orthonormalize them. This is where the Gram-Schmidt process comes in. It takes a set of vectors and turns them into an orthonormal set of vectors (each vector has unit length and is orthogonal to all others).
Here's a step-by-step outline of the process:

• Take your first eigenvector \mathbf{v_1} and normalize it to get \mathbf{u_1}: \[ \mathbf{u_1} = \frac{\mathbf{v_1}}{\|\mathbf{v_1}\|} \]
• Adjust the next eigenvector \mathbf{v_2} to make it orthogonal to \mathbf{u_1} by subtracting the projection of \mathbf{v_2} onto \mathbf{u_1}: \[ \mathbf{u_2} = \frac{\mathbf{v_2} - (\mathbf{u_1} \cdot \mathbf{v_2})\mathbf{u_1}}{\|\mathbf{v_2} - (\mathbf{u_1} \cdot \mathbf{v_2})\mathbf{u_1}\|} \]
• Repeat for the third eigenvector \mathbf{v_3}, ensuring it is orthogonal to both \mathbf{u_1} and \mathbf{u_2}: \[ \mathbf{u_3} = \frac{\mathbf{v_3} - (\mathbf{u_1} \cdot \mathbf{v_3})\mathbf{u_1} - (\mathbf{u_2} \cdot \mathbf{v_3})\mathbf{u_2}}{\|\mathbf{v_3} - (\mathbf{u_1} \cdot \mathbf{v_3})\mathbf{u_1} - (\mathbf{u_2} \cdot \mathbf{v_3})\mathbf{u_2}}\|} \]

After applying this process, we get an orthonormal basis set of eigenvectors.

In linear algebra, an orthonormal basis of a vector space is a set of vectors that are both orthogonal and normalized. This means each vector has a magnitude of one (unit length) and is perpendicular to all other vectors in the set. The importance of an orthonormal basis lies in simplifying many problems:

• They make matrix operations simpler, as dot products become trivial and matrix inversions are straightforward.
• In quantum mechanics and computer graphics, orthonormal bases are crucial for defining spaces and transformations.

To form an orthonormal basis from a set of eigenvectors, we utilize the Gram-Schmidt process. This ensures that all new vectors created are mutually orthogonal and of unit length.
Checking our final orthonormal basis, we should verify:

• Each vector should have a magnitude of one: \[ \|\mathbf{u_i}\| = 1 \]
• Each pair of vectors should be orthogonal: \[ \mathbf{u_i} \cdot \mathbf{u_j} = 0, \text{ for } i \eq j \]

After verification, we can confidently use the orthonormal basis in our solutions and applications.