Get Dynamics Of Tuples Of Matrices In Jordan Form

The given Jordan canonical form implies, minimal polynomial of corresponding matrix should be x2=0. Hence if matrix A is having the property that A≠0 and A2=0, it will have desired 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.

Every matrix is similar to a matrix in Jordan form in Jordan form, being a direct sum of upper triangular matrices, is itself an upper triangular matrix. As such, its diagonal elements are equal to its eigenvalues.

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Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix.

A matrix is said to be in Jordan form if 1) its diagonal entries are equal to its eigenvalues; 2) its supradiagonal entries are either zeros or ones; 3) all its other entries are zeros. We are going to prove that any matrix is equivalent to a matrix in Jordan form.

The dimension of the eigenspace null(T − a) tells you exactly how many Jordan blocks there are, since each Jordan block has a 1-dimensional eigenspace. In other words, t1 is the number of Jordan blocks. If T has only Jordan blocks of size 1, then t2 = dim null(T −a)2 is the same as t1.