A linear transformation $$ T:R^3→R^3$$ is defined as $$ T:(x)=Cx$$ where $C$ is a symmetric matrix.

a) State the dimensions of the eigenspaces $\mbox{N(C-αI)}$ and $N(c-βI)$

It is also given that: $$C\begin{pmatrix} -3 \\ 0 \\ 1 \\ \end{pmatrix} = \begin{pmatrix} -3α \\ 0 \\ α \\ \end{pmatrix}, C\begin{pmatrix} -2 \\ 1 \\ 0 \\ \end{pmatrix} = \begin{pmatrix} 2α \\ α \\ 0 \\ \end{pmatrix} $$

It is given that $C$ has a repeated eigenvalue $α$ and another eigenvalue $β$, where $α \neq β$

b) Find a basis for $\mbox{N(C-αI)}$ and $N(c-βI)$

→I understand that if $C$ is symmetric then $C$ has to be diagonalisable so that if $α$ is repeated the algebraic multiplicity= geometric multiplicity= 2, so $N(C-αI)$ is $2$ dimensional and $N(c-βI)$ is $1$ dimensional.

However, for part b), I don't understand how I would find the bases for the eigenspaces with the information I've been given.

  • $\begingroup$ What is $\beta$? $\endgroup$ – Berci Feb 10 at 12:53
  • $\begingroup$ I had omitted an important sentence from the question, I added it in bold. Other than that I have no other information. $\endgroup$ – Jacopo Salvatore Feb 10 at 13:20

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