We say $\lambda$ is an eigenvalue of a square matrix $A$ if $Ax = \lambda x$.

Now, i want two examples of a matrix like $A$.
The first one, $A$ should have just one eigenvalue which should be $0$.
The second one, $A$ should be a matrix in which $a_1,\dots,a_n$ are eigenvalues. ( They give $a_i$'s and want the matrix )

Note : I have no idea how to find these matrices. I don't know where to start ...

  • $\begingroup$ HINT: Diagonal matrices $\endgroup$ – Andrew Oct 4 '16 at 23:15
  • $\begingroup$ @Andrew for the second one ? $\endgroup$ – Arman Malekzadeh Oct 4 '16 at 23:21

So, with the hint andrew gave me, i solved the problem myself...
If a matrix is diagonal and $a_1,\dots,a_n$ are on the diagonal, then we have equations like :
($a_i-\lambda )(x_i)=0$
And one matrix which has zero as an eigenvalue and zero is its only eigenvalue is the zero matrix itself !!! ( For example, zero matrix in $M_2(\mathbb R)$ )

  • $\begingroup$ Not quite true: Any triangular matrix with $0$s on the diagonal (or more generally any nilpotent matrix), such as the matrix $\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$, has zero as its only eigenvalue. $\endgroup$ – arkeet Oct 4 '16 at 23:36
  • $\begingroup$ @arkeet Thank you :) i made edit $\endgroup$ – Arman Malekzadeh Oct 4 '16 at 23:38

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