Jordan Normal Form - Powers

Powers

If n is a natural number, the nth power of a matrix in Jordan normal form will be a direct sum of upper triangular matrices, as a result of block multiplication. More specifically, after exponentiation each Jordan block will be an upper triangular block.

For example,

\begin{bmatrix} 2 & 1 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 5 & 1 \\ 0 & 0 & 0 & 0 & 5 \end{bmatrix}^4 =\begin{bmatrix} 16 & 32 & 24 & 0 & 0 \\ 0 & 16 & 32 & 0 & 0 \\ 0 & 0 & 16 & 0 & 0 \\ 0 & 0 & 0 & 625 & 500 \\ 0 & 0 & 0 & 0 & 625 \end{bmatrix}.

Further, each triangular block will consist of λn on the main diagonal, times λn-1 on the upper diagonal, and so on. This expression is valid for negative integer powers as well if one extends the notion of the binomial coefficients .

For example,

\begin{bmatrix} \lambda_1 & 1 & 0 & 0 & 0 \\ 0 & \lambda_1 & 1 & 0 & 0 \\ 0 & 0 & \lambda_1 & 0 & 0 \\ 0 & 0 & 0 & \lambda_2 & 1 \\ 0 & 0 & 0 & 0 & \lambda_2 \end{bmatrix}^n =\begin{bmatrix} \lambda_1^n & \tbinom{n}{1}\lambda_1^{n-1} & \tbinom{n}{2}\lambda_1^{n-2} & 0 & 0 \\ 0 & \lambda_1^n & \tbinom{n}{1}\lambda_1^{n-1} & 0 & 0 \\ 0 & 0 & \lambda_1^n & 0 & 0 \\ 0 & 0 & 0 & \lambda_2^n & \tbinom{n}{1}\lambda_2^{n-1} \\ 0 & 0 & 0 & 0 & \lambda_2^n \end{bmatrix}.

Read more about this topic:  Jordan Normal Form

Famous quotes containing the word powers:

    “All the powers of imagination combine in hypochondria.”
    —Mason Cooley (b. 1927)

    “A multitude of causes unknown to former times are now acting with a combined force to blunt the discriminating powers of the mind, and unfitting it for all voluntary exertion to reduce it to a state of almost savage torpor.”
    —William Wordsworth (1770–1850)