For eigen value it needed help
https://gateoverflow.in/400451/%23eigen-vectors
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To find the eigenvalues and eigenvectors of the given matrix using its upper triangular property, we can proceed as follows:
The eigenvalues of an upper triangular matrix are simply its diagonal entries. Therefore, the eigenvalues of the given matrix are 1, 2, and 3.
To find the eigenvectors, we need to solve the equation (A - λI)x = 0 for each eigenvalue λ. Here, I is the identity matrix of the same size as A. For example, to find the eigenvectors corresponding to the eigenvalue λ = 1, we need to solve the equation:
(A - λI)x = (1 - 1) x = 0
Substituting the values of A and λ, we get:
\begin{pmatrix} 1 & 0 & 1 \ 0 & 2 & 3 \ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}
Simplifying this equation, we get:
x1 + x3 = 0 2x2 + 3x3 = 0 3x3 = 0
From the last equation, we can see that x3 = 0. Substituting this value in the first equation, we get x1 = 0. Substituting x3 = 0 in the second equation, we get 2x2 = 0, which implies x2 = 0. Therefore, the eigenvectors corresponding to the eigenvalue λ = 1 are of the form:
x = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}t
where t is any non-zero scalar.
- Similarly, we can find the eigenvectors corresponding to the other eigenvalues λ = 2 and λ = 3. For λ = 2, we need to solve the equation:
(A - λI)x = \begin{pmatrix} -1 & 0 & 1 \ 0 & 0 & 3 \ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}
Simplifying this equation, we get:
-x1 + x3 = 0 3x2 = 0 x3 = 0
From the second equation, we can see that x2 = 0. Substituting this value in the first equation, we get x1 = x3. Therefore, the eigenvectors corresponding to the eigenvalue λ = 2 are of the form:
x = \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}t
where t is any non-zero scalar.
- For λ = 3, we need to solve the equation:
(A - λI)x = \begin{pmatrix} -2 & 0 & 1 \ 0 & -1 & 3 \ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}
Simplifying this equation, we get:
-2x1 + x3 = 0 -x2 +
