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:

    “Exploitation and oppression is not a matter of race. It is the system, the apparatus of world-wide brigandage called imperialism, which made the Powers behave the way they did. I have no illusions on this score, nor do I believe that any Asian nation or African nation, in the same state of dominance, and with the same system of colonial profit-amassing and plunder, would have behaved otherwise.”
    —Han Suyin (b. 1917)

    “And as the sun above the light doth bring,
    Though we behold it in the air below,
    So from th’ eternal Light the soul doth spring,
    Though in the body she her powers do show.”
    —Sir John Davies (1569–1626)

    “Great abilites are not requisite for an Historian; for in historical composition, all the greatest powers of the human mind are quiescent. He has facts ready to his hand; so there is no exercise of invention. Imagination is not required in any degree; only about as much as is used in the lowest kinds of poetry. Some penetration, accuracy, and colouring, will fit a man for the task, if he can give the application which is necessary.”
    —Samuel Johnson (1709–1784)