Question

T(x1,x2,...xn) = (x1, 2x2, 3x3, ...nxn), find all eigenvalues and eigenvectors.

A. Complex numbers only
B. Real numbers only
C. Imaginary numbers only
D. Complex and real numbers

Answer 1

To find the eigenvalues and eigenvectors of the given function, we can solve the equation T(x1,x2,...xn) = λ(x1, 2x2, 3x3, ...nxn). Let's consider an example with n = 3. The eigenvalues are λ = 1 and x1 = 0, and the corresponding eigenvectors are (x1, x2, x3) and (0, x2, x3), where x1, x2, and x3 are real numbers.

Step-by-step explanation:

The given function is T(x1,x2,...xn) = (x1, 2x2, 3x3, ...nxn). To find the eigenvalues and eigenvectors, we need to solve the equation T(x1,x2,...xn) = λ(x1, 2x2, 3x3, ...nxn), where λ is the eigenvalue. Let's consider an example with n = 3.

T(x1, x2, x3) = (x1, 2x2, 3x3)

λ(x1, 2x2, 3x3) = (x1, 2x2, 3x3)

Expanding the equation, we get:

x1 = λx1

2x2 = 2λx2

3x3 = 3λx3

From the first equation, we can see that x1 = 0 or λ = 1.

If x1 = 0, then the second and third equations become 0 = 0, which means x2 and x3 can be any real numbers. So, the eigenvector is (0, x2, x3), where x2 and x3 are real numbers.

If λ = 1, then the second equation becomes 2x2 = 2x2, which means x2 can be any real number. The third equation becomes 3x3 = 3x3, which means x3 can be any real number. So, the eigenvector is (x1, x2, x3), where x1, x2, and x3 are real numbers.