Let the 2x2 matrix How can I found a necessary and sufficient condition in order the matrix A to be similar to the following matrix
I would be grateful if someone show me a step-by-step solution.
Thanks in Advance
SIMILAR MATRICES AND CHANGE OF BASIS
that link should help
and NO Peteryellow its not true
Peteryellow: Please re-read the first post. What is being asked is a necessary and sufficient condition.
ADARSH: Any matrix is similar to itself -in fact similarity is an equivalence relation-, that is what Peteryellow meant ( what I interpret, because he certainly didn't say that),
As for the question:
The 2 matrixes are similar if and only if there exists a non-singular Matrix such that
Or equivalently: and is invertible
Let's work with a generic matrix: with (1) so that it is indeed invertible.
Now:
Equivalently:
Solve for (see under what conditions on a, b, c, d this is possible), and then what other conditions you have to add for (1) to hold. And then you are done.
Here's a solution that cuts out some of the algebra.
If A is similar to then will be similar to .
So suppose that is an invertible matrix such that .
The inverse of P is given by , where is the determinant of P. Then
.
Comparing coefficients, you see that and .
Therefore , , and are not both 0. Conversely, if a,b,c,d satisfy those conditions then you can reconstruct w,x,y,z so that the matrices and are similar. So the conditions in red are the necessary and sufficient conditions for A to be similar to .