Find the eigenvalues and eigenvectors of the following matrices.\(\left(\begin{array}{lll}1 & 2 & 2 \\ 2 & 3 & 0 \\ 2 & 0 & 3\end{array}\right)\)

The characteristic equation is a fundamental concept in linear algebra that helps us find the eigenvalues of a matrix. For any given square matrix \(A\), the characteristic equation is written as \( \text{det}(A - \lambda I) = 0 \). Here, \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix. This equation essentially transforms the matrix into a polynomial of \( \lambda \), called the characteristic polynomial.

In simpler terms, by solving this polynomial, we can discover the eigenvalues of the matrix. The solutions to this equation are the values of \( \lambda \) that make the determinant equal to zero. These eigenvalues are crucial as they provide insights into many properties of the matrix, like its stability and invertibility.

To illustrate, consider the matrix \( A \) from our problem: \[ \begin{pmatrix} 1 & 2 & 2 \ 2 & 3 & 0 \ 2 & 0 & 3 \ \end{pmatrix} \] We write the characteristic equation as: \[ \text{det} \begin{pmatrix} 1-\lambda & 2 & 2 \ 2 & 3-\lambda & 0 \ 2 & 0 & 3-\lambda \end{pmatrix} = 0 \]

Solving the determinant of this 3x3 matrix, we can derive the characteristic polynomial, and ultimately, the eigenvalues.

• Write the matrix \( A \)
• Form \( A - \lambda I \)
• Compute the determinant and set it to zero

By understanding and solving the characteristic equation, finding eigenvalues becomes a straightforward process in linear algebra.

The determinant of a matrix is a special number that can tell us a lot about the matrix. It is a scalar value derived from a square matrix and helps determine whether the matrix is invertible, among other properties.

To compute the determinant, we often use a method known as cofactor expansion. For a 3x3 matrix, the formula looks like this: \[ \text{det}(A) = a(ei − fh) – b(di − fg) + c(dh − eg) \] where the matrix \( A \) is:
\[ \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \ \end{pmatrix} \]

In our example, we need to find the determinant of \( A - \lambda I \), resulting in the following matrix: \[ \begin{pmatrix} 1-\lambda & 2 & 2 \ 2 & 3-\lambda & 0 \ 2 & 0 & 3-\lambda \end{pmatrix} \]

We compute this determinant as follows:

• Expand along the first row
• Find the 2x2 determinants of the resulting submatrices
• Combine those results using the cofactor expansion formula

After performing these calculations, the determinant simplifies to the characteristic polynomial \( \lambda^3 - 10\lambda + 20\lambda - 12 = 0 \).

The determinant not only helps in finding eigenvalues but also in understanding the matrix's properties generally used in solving linear systems and other areas of linear algebra.

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It provides powerful tools for various applications in engineering, physics, computer science, and economics.

One of the key concepts in linear algebra is matrix operations, including addition, multiplication, and finding inverses. Matrices represent linear transformations and can form systems of linear equations.

• Vectors and vector spaces
• Matrix operations
• Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are particularly important because they characterize linear transformations. They provide information about how a transformation changes the space it acts upon. If \( A \) is a square matrix, an eigenvector \( x \) and a corresponding eigenvalue \( \lambda \) satisfy the equation \( A\vec{x} = \lambda\vec{x} \).

For the matrix problem given: \[ A = \begin{pmatrix} 1 & 2 & 2 \ 2 & 3 & 0 \ 2 & 0 & 3 \ \end{pmatrix} \]

By solving the characteristic equation \( \text{det}(A − \lambda I) = 0 \) and finding the eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \), we get the eigenvectors corresponding to each eigenvalue. These vectors show the direction that does not change under the transformation represented by the matrix.

Linear algebra, with its deep theoretical constructs and practical applications, forms the backbone of many advanced mathematical, scientific, and engineering tasks. Knowing how to compute and interpret eigenvalues and eigenvectors enhances our understanding of various phenomena modeled by matrices.