Equivalent matrix: changebase

### 5.47.8 Equivalent matrix: changebase

The changebase command changes a matrix to represent the same linear function in a different basis.

- changebase takes two arguments:
  - A, a matrix.
  - P, a change-of-basis matrix.
- changebase(A,P) returns the matrix P−1AP.

#### Examples

- Input:
  changebase([[1,2],[3,4]],[[1,0],[0,1]])
  Output:

  ⎡ ⎢ ⎣ |
  1 | 2
  3 | 4
  | ⎤ ⎥ ⎦

- Input:
  changebase([[1,1],[0,1]],[[1,2],[3,4]])
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
  −5 | −8
  9
  2
  | 7
  | ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

  Indeed:

  ⎡ ⎢ ⎣ |
  1 | 2
  3 | 4
  | ⎤ ⎥ ⎦ |
  −1
  | · | ⎡ ⎢ ⎣ |
  1 | 1
  0 | 1
  | ⎤ ⎥ ⎦ | · | ⎡ ⎢ ⎣ |
  1 | 2
  3 | 4
  | ⎤ ⎥ ⎦ | = | ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
  −5 | −8
  9
  2
  | 7
  | ⎤ ⎥ ⎥ ⎥ ⎥ ⎦