If two ℝeal valued, non-identity matrices A (dimension MxN) and B (dimension NxN)(adsbygoogle = window.adsbygoogle || []).push({});

satisfy the condition A * B = A

Is there a name for the relationship between B and A ?

For example for: [itex]A =

\left( \begin{array}{cc}

\frac{1}{3} & \frac{2}{3} \end{array} \right), \quad B = \left( \begin{array}{cc}

\frac{3}{5} & \frac{2}{5} \\

\frac{1}{5} & \frac{4}{5} \end{array} \right) \quad A * B = \left( \begin{array}{cc}

\frac{1}{3} & \frac{2}{3} \end{array} \right)

[/itex]

My Observations:

- If such A exists for B, then for any real number
k, k*A is also a solution for B

since (k*A)*B = k*(A*B)=k*A

implies, there will be infinite solutions for B.

.- [itex]

p\to+\infty{\left( \begin{array}{cc}

\frac{3}{5} & \frac{2}{5} \\

\frac{1}{5} & \frac{4}{5} \end{array} \right)}^p = \left( \begin{array}{cc}

\frac{1}{3} & \frac{2}{3} \\

\frac{1}{3} & \frac{2}{3} \end{array} \right)

[/itex]

My Question: Given a matrix B, how do we find A such that A*B=A ?

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# Matrix Question: A*B=A (B is non-identity)

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