Do similar matrices have the same Jordan canonical form?

Less abstractly, one can speak of the Jordan canonical form of a square matrix; every square matrix is similar to a unique matrix in Jordan canonical form, since similar matrices correspond to representations of the same linear transformation with respect to different bases, by the change of basis theorem.

How do you find the Jordan canonical form of a matrix?

be the n by r matrix whose columns are these vectors in reverse order. Once we’ve done this for all eigenvalues λ stick the matrices Pλ together horizontally to get an n by n matrix P. Then P will be non-singular, and P−1AP = J, the Jordan form.

Do similar matrices have the same determinant?

Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues. It is often important to select a matrix similar to a given one but having a possibly simpler form, for example, diagonal form (see Diagonal matrix) or Jordan form (see Jordan matrix).

How do you prove similar matrices have the same eigenvalues?

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

How can you tell if two matrices are similar?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors). When we diagonalize A, we’re finding a diagonal matrix A that is similar to A.

Why do we need Jordan canonical form?

Jordan form is also important for determining whether two matrices are similar. In particular, we can say that two matrices will be similar if they “have the same Jordan form”. Exercise: using Jordan canonical form, prove that a matrix is diagonalizable (over C) iff the minimal polynomial has no repeated roots.

What is controllable canonical form?

1.1 Controllable Canonical Form. The controllable canonical form arranges the coefficients of the transfer func- tion denominator across one row of the A matrix:  

How do you know if two matrices are similar?

If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors).

How do you prove two matrices are similar?

Definition (Similar Matrices) Suppose A and B are two square matrices of size n . Then A and B are similar if there exists a nonsingular matrix of size n , S , such that A=S−1BS A = S − 1 B S .

Can a matrix be similar to a Jordan matrix?

Two matrices may have the same eigenvalues and the same number of eigen­ vectors, but if their Jordan blocks are different sizes those matrices can not be similar. Jordan’s theorem says that every square matrix A is similar to a Jordan matrix J, with Jordan blocks on the diagonal: ⎡ ⎤ J = ⎢ ⎢ ⎢ ⎣ J10 ··· 0 0 J2··· 0 . . . . 0 0 ··· J d

How are the eigenvalues in a Jordan matrix?

In a Jordan matrix, the eigenvalues are on the diagonal and there may be ones above the diagonal; the rest of the entries are zero. The number of blocks is the number of eigenvectors – there is one eigenvector per block. To summarize: • If A has n distinct eigenvalues, it is diagonalizable and its Jordan matrix is the diagonal matrix J = Λ.

How is the number of blocks related to the Jordan matrix?

The number of blocks is the number of eigenvectors – there is one eigenvector per block. To summarize: • If A has n distinct eigenvalues, it is diagonalizable and its Jordan matrix is the diagonal matrix J = Λ. • If A has repeated eigenvalues and “missing” eigenvectors, then its Jordan matrix will have n − d ones above the diagonal.

Which is an example of a Jordan form?

Jordan form Camille Jordan found a way to choose a “most diagonal” representative from each family of similar matrices; this representative is said to be in Jordan nor­ 4 1 4 0 mal form. For example, both 0 4 and 0 4 are in Jordan form. This form used to be the climax of linear algebra, but not any more.

https://www.youtube.com/watch?v=GVixvieNnyc