CharPoly(M) = Characteristic polynomial of square matrix or scalar M


CharPoly(M) = Characteristic polynomial of square matrix or scalar M:

The characteristic polynomial is the polynomial left-hand side of the characteristic equation:

Det(M − x * I) = 0

Where M is an n by n square matrix and I is the identity matrix of the same size. The roots of the characteristic polynomial are the eigenvalues of the matrix.

Det(M − x * I) = P(x) = p(n)*x^n + p(n−1)*x^(n−1) + ... + p(1)*x + p(0) and can be represented as an (n + 1) by 1 column vector {p(n); p(n−1); ..., p(1); p(0)}.

See: Characteristic Polynomial -- From MathWorld
See: Characteristic Polynomial -- From Jean-Pierre Moreau's Home Page

If the argument is a scalar s = s.r + s.i * i, where i = SqRt(−1):
If s.i = 0, CharPoly(s) = x − s = {1; −s}
Else CharPoly(s) = x^2 − 2*s.r*x + (s.r^2 + s.i^2) = {1; −2*s.r; s.r^2 + s.i^2}.

For scalars the characteristic polynomial is the unique monic polynomial with real coefficients and least power whose value at s is equals to zero. For example: CharPoly(i) = x^2 + 1 = {1; 0; 1}.

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This page accessed times since September 15, 2006.
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Changes last made on Monday, 06-Aug-07 20:22:39 PDT

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