Question

1. Any 2×2 matrix is similar to one of the following canonical forms: [λ 0 / 0 μ ] , [ α β / -β α] [λ 1 / 0 λ] For each of the following systems x'=Ax: ​Find the eigenvalues and eigenvectors for the given matrix A, expressing them in terms of the parameters or constants present in the matrix.

Answer 1

In linear algebra, a 2x2 matrix can be transformed into canonical forms such as [λ 0 / 0 μ] or [α β / -β α] by applying a similarity transformation. To find the eigenvalues and eigenvectors of a matrix A, we need to solve the equation Ax = λx, where x is the eigenvector and λ is the eigenvalue.

Step-by-step explanation:

In linear algebra, a 2x2 matrix can be transformed into canonical forms such as [λ 0 / 0 μ] or [α β / -β α] by applying a similarity transformation. To find the eigenvalues and eigenvectors of a matrix A, we need to solve the equation Ax = λx, where x is the eigenvector and λ is the eigenvalue. Let's work through an example:

Given matrix A = [[a, b], [c, d]], we can find the eigenvalues by solving the characteristic equation det(A - λI) = 0. For a 2x2 matrix, this equation simplifies to λ² - (a + d)λ + (ad - bc) = 0. Solving this quadratic equation gives us the eigenvalues λ₁ and λ₂.

To find the eigenvectors, we substitute each eigenvalue back into the equation Ax = λx and solve for x. For each eigenvalue, we get a system of linear equations which can be solved to find the eigenvectors.