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,
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,
Read more about this topic: Jordan Normal Form
Famous quotes containing the word powers:
“Religion differs from magic in that it is not concerned with control or manipulation of the powers confronted. Rather it means submission to, trust in, and adoration of, what is apprehended as the divine nature of ultimate reality.”
—Joachim Wach (18981955)
“Great Powers of falling wave and wind and windy fire,
With your harmonious choir
Encircle her I love and sing her into peace,
That my old care may cease....”
—William Butler Yeats (18651939)
“The Federal Constitution has stood the test of more than a hundred years in supplying the powers that have been needed to make the Central Government as strong as it ought to be, and with this movement toward uniform legislation and agreements between the States I do not see why the Constitution may not serve our people always.”
—William Howard Taft (18571930)