Explain why the eigenvalues of triangular and diagonal matri

Explain why the eigenvalues of triangular and diagonal matrices T = (t_11 t_12 ... t_1n 0 t_22 ... t_2n 0 0 ... t_nn) and D = (lambda_1 0 ... 0 0 lambda_2 ... 0 0 0 ... lambda_n) are simply the diagonal entries-the t_ii\'s and lambda_i\'s.

Solution

to find the eigen values (k) of the matrix \'T \" consider the determinant

|T - kI | = | t₁₁ -k   t₁₂ ----- t_1n

0 t22-k ---- t_2n

-------

------------------- -t_nn-k |

expanding along the the 1 st column we get

   ( t₁₁ -k) | t_{22 -k --------} t_2n

0 t₃₃-k --- --------t_3n

   -----

   t_nn-k| once again expanding along the 1 st column

   |T-kI | = (t₁₁-k) ( t₂₂-k) | t₃₃-k ------

   ----

   --------- t_nn-k|

when we proceed like this after n th step we get the charecteristic eqn as

   (t₁₁-k) (t₂₂-k) ----(t_nn-k) =0

and the factors equal to 0 will give the eigen values as k = t₁₁ , t₂₂ ----- t_nn   (only the main diogonal elements

  B .   

let k be the eigen values then the charateristic eqn is given by

   |D-kI |= 0

   | t₁-k 0 -- 0

   0 t₂-k -- 0

   ----- 0

   0 ------------ t_n-k | =0

   each time expanding along the 1 st column we get ( t₁-k) (t₂-k) ---(t_n-k) =0 = > k= t₁,t₂....t_n  the diogonal elements