Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.We start with a field K (you can think of K as the real or complex numbers) and an n-by-n matrix A over K. The characteristic polynomial of A, denoted by pA(t), is the element of the polynomial ring K[t] defined by
- pA(t) = det(A - tI)
The degree of the polynomial pA(t) is n. The most important fact about the characteristic polynomial is this: the eigenvalues of A are precisely the zeros of pA(t). The constant coefficient pA(0) is equal to the determinant of A, and the coefficient of tn-1 is equal to (-1)n-1 times the trace of A.
The Cayley-Hamilton theorem states that replacing t by A in the expression for pA(t) yields the zero matrix: pA(A) = 0. Simply, every matrix satisfies its own characteristic equation. As a consequence of this, one can show that the minimal polynomial of A divides the characteristic polynomial of A.
The matrix A and its transpose have the same characteristic polynomial. If A and B are similar matrices, then they also have the same characteristic polynomial. The converse however isn't true: matrices with the same characteristic polynomial need not be similar.
A is similar to a triangular matrix[?] if and only if its characteristic polynomial can be completely factored into linear factors over K. In fact, A is even similar to a matrix in Jordan normal form[?] in this case.
John,--Hugh's favorite chapter, the fourteenth. A hymn.html">hymn
the morning," and then they all knelt in prayer.html">prayer, the fervent prayer
of all is dead.
Mr. Leslie took no part in the services; he stood with Sibyl as one of
come from New York immediately upon hearing of the accident. Tom and
said; she only wanted to stay by Hugh until the last. So they let her
her God.
Then came another hymn, and slowly the bearers lifted all that was
flower-arch, and down.html">down the garden-walk, where the throng made way for
The darkness deepens; Lord, with me abide;
Help of the helpless, Oh abide with me.
I fear no foe with Thee at hand to bless,
Where is death's sting, where, grave.html">grave, thy victory?
CHAPTER XI.
CONCLUSION.
time upon Hugh's grave. Aunt Faith and Bessie were in the sitting.html">sitting-room
the open hall-door from the back garden, where they were sitting under
growing around it from the little bracket which he had made that last
vivid realistic expression; Bessie saw it, and laying down her work,
not, Aunt Faith?" she said at last.
Aunt Faith put on her glasses, and drew nearer the easel. "It is
came you to succeed so well?"
"I had been working at it all summer, aunt, but the eyes I could not
day at home; don't you remember how I begged for the morning? He.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|