- #1
math8
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Consider the matrix [tex]A = u v^{\ast }[/tex] where [tex]u, v \in \textbf{C}^{n}[/tex]. Under what condition on u and v is A a projector?A is a projector if [tex]A^{2}=A [/tex], so we have [tex]u v^{*} u v^{*}= u v^{\ast }[/tex].
Does this imply [tex] u v^{\ast } = I[/tex] ? And what exactly are the conditions on u and v that they are asking?
do we have that [tex]u_{i} v^{\ast }_{i}=1[/tex] and [tex]u_{i} v^{\ast }_{j}=0[/tex] for [tex] i\neq j[/tex] ?
Does this imply [tex] u v^{\ast } = I[/tex] ? And what exactly are the conditions on u and v that they are asking?
do we have that [tex]u_{i} v^{\ast }_{i}=1[/tex] and [tex]u_{i} v^{\ast }_{j}=0[/tex] for [tex] i\neq j[/tex] ?