Similar matrix

See also: Reduction of endomorphism

In Mathématiques, two square matrices has and B is known as similar if there exists a invertible Matrice P such as:

A = PBP^ {- 1} .

It is about a Relation of equivalence.

Two matrices are similar if and only if they constitute two matrices representative of same the Endomorphisme in two different bases. One should not confuse the concept of similar matrices with that of equivalent Matrices. On the other hand, if two matrices are similar, then they are equivalent. A means of determining if two matrices are similar is them to reduce, i.e. to bring back them to a standard form: diagonal, reduced form of Jordan…

Example

The following matrices are similar:
A = \ begin {pmatrix} 2 & 1 \ \ -2 & 0 \ end {pmatrix}
B = \ begin {pmatrix} -2 & -2 \ \ 5 & 4 \ end {pmatrix} .

Indeed, while posing:

P = \ begin {pmatrix} 1 & 1 \ \ 1 & 0 \ end {pmatrix} ,
one obtains: P^ {- 1} = \ begin {pmatrix} 0 & 1 \ \ 1 & -1 \ end {pmatrix} ,

and it is checked easily that:

P^ {- 1} HAS P = \ begin {pmatrix} -2 & -2 \ \ 5 & 4 \ end {pmatrix} = B.

Invariants of similarity

The applications of the space of the square matrices whose result is identical for a matrix and a matrix which is similar for him are called invariants of similarity . In particular, the Trace of a matrix is an invariant of similarity. With the preceding notations:
\mathrm{tr}\; B = \ mathrm {tr} \ left ((P^ {- 1} A) P \ right) = \ mathrm {tr} \ left (P (P^ {- 1} A) \ right) = \ mathrm {tr} \; A

In the same way, the row, the Determinant, the eigenvalues, the characteristic Polynomial and the minimal polynomial are invariants of similarities.

See too

  • equivalent Matrices

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