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recently I've come across a few cases where I had to evaluate $e^{At}$ (differential equations) where A is in Frobenius canonical form. The algorithm looks like: $$e^{At} = Pe^{Jt}P^{-1}$$ so I need to do Jordan decomposition. From Frobenius canonical form I can easily write the characteristic equation => compute the roots which are eigenvalues and write the Jordan canonical form.

This was the easy part. Now the other part: compute matrices $P$ and $P^{-1}$.

Is there any standard form of those matrices if matrix A is in a Frobenius canonical form?

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  • $\begingroup$ You might be able to use the Frobenius form directly. If the Frobenius form is A=[ 0,1; 1,1 ] with char poly xx-x-1, then xx = x+1 and exp(At) = 1 + At + (At)^2/2 + (At)^3/6 + ... = 1 + At + (A+1)/2 * t^2 + (2A+1)/6 * t^2 + ... = (1 + 1/2 t^2 + 1/6 t^2 + 2/24 t^3 + ...) + A*(t+t^2/2 + t^3/3 + t^4/8 + ... ). The coefficients in the terms are just Fibonacci numbers over factorials. So you get one matrix times a power series in t plus another matrix times a power series in t. If your matrix is n×n, then you get at most n matrices times n power series. $\endgroup$ Feb 21, 2011 at 13:31
  • $\begingroup$ This is a neat problem. I think your idea to use P directly turns out to be better than my simple recurrence. P is a very natural and nice matrix. It is probably as stable as one could expect given that one is working with JCF at all. $\endgroup$ Feb 22, 2011 at 0:27

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The matrix P is the matrix of coefficients of certain divisors of the characteristic polynomial of the Frobenius canonical form.

In fact this works for any companion matrix, not just the companion matrix of a power of an irreducible polynomial.

Suppose a matrix A has 1s above the diagonal, and the last row is [ v0, v1, v2, v3, ..., vn ], so $$ A = \begin{bmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 0 & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & 1 & 0 \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ v_0 & v_1 & v_2 & v_3 & \ldots & v_{n-1} & v_n \\ \end{bmatrix}.$$ Then $$ A^{(n+1)} = v_0 A^0 + v_1 A^1 + v_2 A^2 + \ldots + v_n A^n$$

Suppose $x-\alpha$ is a factor of $$ f(x) = x^{(n+1)} - v_n x^n - \ldots - v_1 x^1 - v_0 x^0 $$ and write $$ f(x)/(x-\alpha) = w_n x^n + w_{n-1} x^{(n-1)} + \ldots + w_1 x^1 + w_0 x^0 $$ where of course $w_n = 1$. Then $w = [ w_0, w_1, w_2, \ldots, w_n ]$ is a left eigenvector of A, since $wA = \alpha w$.

In more detail, $$wA = [ v_0, w_0 + v_1, w_1 + v_2, \ldots, w_{n-1} + v_n ]$$ and $$f(x) = (x-\alpha)( w_n x^n + w_{n-1} x^{(n-1)} + \ldots + w_1 x^1 + w_0 x^0) = x^{n+1} - (-\alpha+w_{n-1}) x^{n} \ldots$$ so by equating coefficients of $x^n$, one gets $v_n = \alpha - w_{n-1}$ and $v_n + w_{n-1} = \alpha w_n$. Similarly, $v_{n-1} = \alpha w_{n-1} - w_{n-2}$, and in general $v_i = \alpha w_i - w_{i-1}$ so $v_i + w_{i-1} = \alpha w_i$ and $wA = \alpha w$.

Something similar works with repeated roots even, to give you Jordan blocks.

So to find your matrix P, find each root α of f(x) with multiplicity k, and then the rows of P are the coefficients (in increasing order of power of x) of f(x)/(x−α)i, from i=1 to k, with corresponding diagonal entry of the matrix just being α, and the run from 1 to k forming the Jordan block.


Example 1

Find the JCF of $$A = \begin{bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-1&4&-6&4\end{bmatrix}$$

The characteristic polynomial is just $f(x) = x^4-4x^3+6x^2-4x+1$ since it is a companion matrix of $f$. Of course, $f(x) = (x-1)^4$ is easy to factor.

We calculate $$f(x)/(x-1) = 1x^3 - 3x^2 + 3x - 1,$$ $$f(x)/(x-1)^2 = 0x^3+1x^2-2x+1,$$ $$f(x)/(x-1)^3 = 0x^3+0x^2+x-1,\text{ and}$$ $$f(x)/(x-1)^4 = 0x^3+0x^2+0x+1.$$

We then write down the coefficients (in reverse order, I guess): $$P = \begin{bmatrix} -1 & 3 & -3 & 1 \\ 1 & -2 & 1 & 0 \\ -1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}$$ so that $$ P A P^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$$

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    $\begingroup$ Wow. Great answer! Thank you. $\endgroup$
    – kubal5003
    Feb 22, 2011 at 7:56

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