Linear Algebra

Question

We know,

the eigen value for upper triangular/lower triangular/diagonal matrices are the diagonal elements of the matrix.

https://gateoverflow.in/858/gate2002-5a

This question,

https://gateoverflow.in/1174/gate2005-49

If apply row transformation to convert it into upper triangular matrix then my eigen values will be diagonals, but the result is not correct. I know for determinant, applying row transformation or column transformation value of the determinant remains the same.

Why can't we make the given matrix to upper triangular matrix, so that eigen values will be equal to the elements in the diagonals?

 

Comments

the eigen value for upper triangular/lower triangular/diagonal matrices are the diagonal elements of the matrix.

it is true.

 

I know for determinant, applying row transformation or column transformation value of the determinant remains the same.

if R_i ---> k.R_i , then determinant is changed.

if R_i ---> R_i + (k.R_j) , then Determinant is not changed.

 

Note that, if A and B , Determinants is equal, then Eigen values are need not to be same for A and B.

$\begin{pmatrix} 2 & -1\\ -4 & 5 \end{pmatrix}$   ---------> (i)

after applying row transformation R₁ ---> R₁ + ($\frac{1}{5}$R₂)

$\begin{pmatrix} \frac{6}{5} & 0\\ -4 & 5 \end{pmatrix}$  ---------> (ii)

(i) and (ii) have determinant value same(=6),

but eigen values of (i) is 1,6 and eigen values of (ii) is  $\frac{6}{5} $,5 which are different.

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thanks bro