require(pracma)## Loading required package: pracmaB <- matrix(c(-12, 30, -5, 13), nrow = 2, byrow = TRUE)
c <- eigen(B)\[B = \left[\begin{array} {rr} -12 & 30 \\ -5 & 13 \end{array}\right] \]
Values:
3, -2
Vectors:
-0.8944272, -0.4472136, -0.9486833, -0.3162278
Because each value appears only once in the eigen values above both 3 and -2 have a multiplicity of 1.
\[B - \lambda I_2\] =
m1 <- matrix(B + 2*diag(2), nrow=2)
m2 <- matrix(B - 3*diag(2), nrow=2)\[B = \left[\begin{array} {rr} -12 & 30 \\ -5 & 13 \end{array}\right] \]
-10, -5, 30, 15
\[\left[\begin{array} {rr} -10 & 30 \\ -5 & 15 \end{array}\right] \]
1, 0, -3, 0
\[\left[\begin{array} {rr} 1 & -3 \\ 0 & 0 \end{array}\right] \]
In reduced row from there is s single dimension of non-zero rows so the eigenspace of -2 is one
\[B = \left[\begin{array} {rr} -12 & 30 \\ -5 & 13 \end{array}\right] \]
-15, -5, 30, 10
\[\left[\begin{array} {rr} 15 & 30 \\ -5 & 10 \end{array}\right] \]
1, 0, -2, 0
\[\left[\begin{array} {rr} 1 & -2 \\ 0 & 0 \end{array}\right] \]
In reduced row from there is s single dimension of non-zero rows so the eigenspace of 3 is one
Note to all…feel free to make this more clear, or tell me I am missing something…I am struggling immensely with the heavy symbolic and computational nature of learning something that is not being tied to anything real I can understand it by. If this is not right, I will never see it until it reflects real events in the real world…numbers alone are just letters in a box, no words, no communication!