Linear algebra Eigens

Question

Find a 2x2 matrix A for which E_-5 = span {[-3,1]} , E_-4 = span {[5,-2]}

Where E_λ denotes the eigenspace associated with the eigenvalue λ.

Answer 1

Let A = | a b |

| c d |.

Then we have that λI₂ - A

= | λ - a -b |

| -c λ - d |

and det(λI₂ - A ) = (λ - a)(λ - d) - bc = λ² - (a + d)λ + ad - bc. Since we are given that -5 and -4 are eigenvalues for A, we also know that det(λI₂ - A ) = 0 if and only if λ² - (a + d)λ + ad - bc = (λ - (-5))(λ - (-4)) = (λ + 5)(λ + 4) = λ² + 9λ + 20. This further implies that a + d = -9 and ad - bc = 20, which also gives us that d = -9 - a, a(-9-a) - bc = 20, -9a - a² - bc = 20, and a² + 9a + 20 + bc = 0.

Since -5I₂ - A = | -5 - a -b |

| -c -5 - d |

= | -5 - a -b |

| -c -5 - (-9 - a) |

= | -5 - a -b |

| -c 4 + a |

and [-3 1] is an eigenvector associated with the eigenvalue -5, we also know that (-5 - a)(-3) -b(1) = 15 + 3a - b = 0 and -c(-3) + (4 + a)(1) = 3c + 4 + a = 0. Since the two preceding equations are equivalent to 3a - b = -15 and a + 3c = -4, we can deduce by multiplying the equation a + 3c = -4 by -3 and adding to the equation 3a - b = -15 that -9c - b = -3 and b + 9c = 3.

Since -4I₂ - A = | -4 - a -b |

| -c -4 - d |

= | -4 - a -b |

| -c -4 - (-9 - a) |

= | -4 - a -b |

| -c 5 + a |

and [5 -2] is an eigenvector associated with the eigenvalue -4, we also know that (-4 - a)(5) - b(-2) = -20 - 5a + 2b = 0 and -c(5) + (5 + a)(-2) = -5c - 10 - 2a = 0. Since the two preceding equations are equivalent to 5a - 2b = -20 and 2a + 5c = 10, we can deduce by multiplying the equation 5a - 2b = -20 by -2 and adding to 5 times the equation 2a + 5c = 10 that 4b + 25c = 90.

Since we know from the preceding work that b + 9c = 3 and 4b + 25c = 90, we can also conclude by multiplying the equation b + 9c = 3 by -4 and adding to the equation 4b + 25c = 90 that -11c = 78 and c = 78/-11 = -78/11. We can further conclude that b = 3 - 9(-78/11) = 3 + 702/11 = 735/11. Since we also know from the preceding work that a + 3c = -4, we can conclude that a = -4 - 3(-78/11) = -4 + 234/11 = 190/11. We can further conclude from the equation d = -9 - a that d = -9 -(190/11) = -289/11.

Finally, we can conclude that the requested 2 x 2 matrix A is

| 190/11 735/11|

| -78/11 -289/11|.