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Suppose $A$ is a $n \times n$ matrix from $M_n(\mathbb{C})$ with eigenvalues $\lambda_1, \ldots, \lambda_s$. Let $$m(\lambda) = (\lambda - \lambda_1)^{m_1} \ldots (\lambda - \lambda_s)^{m_s}$$

be the minimal polynomial of $A$.

We define $f(A)$ in general to be $p(A)$ for some polynomial interpolating $f(\lambda_k), f'(\lambda_k), \ldots, f^{(m_k - 1)}(\lambda_k)$ for all $k$, assuming that $f$ is defined at these points. This is well defined because for any such two interpolating polynomials $p$ and $q$ we have $p(A) = q(A)$ and such a polynomial always exists. We can also find a polynomial $p$ which has degree less than the minimal polynomial $m$.

Also, then there exist linearly independent matrices $Z_{ij}$ such that for any $f$ defined at $f(\lambda_k), f'(\lambda_k), \ldots, f^{(m_k - 1)}(\lambda_k)$ for all $k$, we have

$$f(A) = \sum_{k = 1}^{s} ( f(\lambda_k)Z_{k0} + f'(\lambda_k)Z_{k1} + \ldots + f^{(m_k - 1)}(\lambda_k)Z_{k, m_k - 1})$$


Okay, with the definitions and such out of the way here's the actual question..

We can use this definition to calculate $f(A)$ for $f$ such as $f(x) = 1/x$ which gives us the inverse of $A$, or $f(x) = \sqrt{x}$ which gives us the square root of $A$, or $f(A)$ for any polynomial $f(x)$. The power series definition of $\sin(A)$, $\cos(A)$, etc also agrees with the definition.

Now this is all really awesome and interesting in my opinion, but why on earth does this work? I have no idea why the properties of the inverse function $1/x$ or the square root $\sqrt{x}$ are "passed on" to matrices this way.

Also, what is this called? Spectral something, but I'm not sure. I tried looking it up on Wikipedia and Google but couldn't find anything. Besides answers to help me understand what this is all about, I'll appreciate any references, keywords and so on.

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+1 very interesting question –  dato datuashvili Sep 1 '12 at 18:35
    
I do not get your second line - how could you define $f(A)$ to be $p(A)$ interpolating the derivatives of $f(A)$? What do you really mean? –  Bombyx mori Sep 1 '12 at 19:03
    
The keyword here is "functional calculus." –  Qiaochu Yuan Sep 1 '12 at 21:28
2  
In addition to the excellent answer you got from Agustí, have a look at the discussion in Higham's book, as well as this nice article. –  J. M. Sep 2 '12 at 3:10

2 Answers 2

up vote 7 down vote accepted

Two classic books where you can find all the details about this stuff:

  1. Gantmacher, Matrix theory, Chelsea.
  2. Lancaster-Titsmenesky, The theory of matrices, Academic Press.

Actually, for "hand" calculations, this works through the Jordan canonical form: you find the Jordan canonical form of your matrix, together with the change of basis matrix

$$ A = SJS^{-1} \ . $$

Then you prove that, for any polynomial $p(t)$, you have

$$ p(A) = S p(J) S^{-1} \ . $$

Hence,

$$ f(A) = S f(J) S^{-1} $$

and you only need to compute $p(J)$ for Jordan matrices.

Which you do as follows: first, if you have a diagonal block matrix

$$ J = \begin{pmatrix} J_1 & 0 & \dots & 0 \\ 0 & J_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & J_r \end{pmatrix} $$

you can easily prove that

$$ p(J) = \begin{pmatrix} p(J_1) & 0 & \dots & 0 \\ 0 & p(J_2) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & p(J_r) \end{pmatrix} $$

So again, on one hand,

$$ f(J) = \begin{pmatrix} f(J_1) & 0 & \dots & 0 \\ 0 & f(J_2) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & f(J_r) \end{pmatrix} $$

and, on the other hand, you just need to know $p(J)$ when $J$ is a Jordan block. If :

$$ J = \begin{pmatrix} \lambda & 0 & 0 & \dots & 0 & 0 \\ 1 & \lambda & 0 & \dots & 0 & 0 \\ 0 & 1 & \lambda & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & \lambda & 0 \\ 0 & 0 & \dots & 0 & 1 & \lambda \end{pmatrix} $$

is an $r\times r$ Jordan block, then

$$ p(J) = \begin{pmatrix} p(\lambda ) & 0 & 0 & \dots & 0 & 0 \\ p'(\lambda)/ 1! & p(\lambda) & 0 & \dots & 0 & 0 \\ p''(\lambda)/ 2! & p'(\lambda)/ 1! & p(\lambda) & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ p^{(r-2)}(\lambda)/(r-2)! &p^{(r-3)}(\lambda)/(r-3)! & \dots & p'(\lambda)/ 1! & p(\lambda) & 0 \\ p^{(r-1)}(\lambda)/(r-1)! &p^{(r-2)}(\lambda)/(r-2)! & \dots & p''(\lambda)/2! & p'(\lambda)/ 1! & p(\lambda) \end{pmatrix} $$

Hence, again you have all in terms of $f$ in fact:

$$ f(J) = \begin{pmatrix} f(\lambda ) & 0 & 0 & \dots & 0 & 0 \\ f'(\lambda)/ 1! & f(\lambda) & 0 & \dots & 0 & 0 \\ f''(\lambda)/ 2! & f'(\lambda)/ 1! & f(\lambda) & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ f^{(r-2)}(\lambda)/(r-2)! &f^{(r-3)}(\lambda)/(r-3)! & \dots & f'(\lambda)/ 1! & f(\lambda) & 0 \\ f^{(r-1)}(\lambda)/(r-1)! &f^{(r-2)}(\lambda)/(r-2)! & \dots & f''(\lambda)/2! & f'(\lambda)/ 1! & f(\lambda) \end{pmatrix} $$

And, in this version of the story, you actually don't need to know your polynomial $p(t)$ for your function $f(t)$ and matrix $A$ -but it's not difficult to find it, anyway: it's called the Lagrange-Sylvester polynomial, which is some sort of mixture between the classic Lagrange interpolation polynomial and a Taylor series.

EDIT

Nevertheless, seems that I forgot to answer the most important question: "Why does this whole thing actually work?"

That is, why defining

$$ f(A) = p(A) $$

for some polynomial $p(t)$ that agrees with $f(t)$ on the spectrum of $A$ all this makes sense? That is, for what reason can we actually call $p(A)$ (the computable thing) the "value" of $f(t)$ on the matrix $A$?

Because of the following:

Theorem. (Gantmacher, chapter V, $\S 4$, theorem 2). If the function $f(t)$ can be expanded in a power series in the circle $\vert t - \lambda \vert < r$,

$$ f(t) = \sum_{n=0}^\infty a_n(t-\lambda)^n \ , $$

then this expansion remains valid when the scalar argument $t$ is replaced by a matrix $A$ whose eigenvalues lie within the circle of convergence.

That is, under the hypotheses of the theorem, you have

$$ f(A) = \sum_{n=0}^\infty a_n(A-\lambda I)^n \ , $$

where the $f(A)$ in the left hand-side means $p(A)$, the value of the Lagrange-Sylvester polynomial on $A$.

So, why not define $f(A)$ as this last power series (i.e., the Taylor series of $f(t)$)? Well, because then you would have to talk a long time about convergence issues of matrix series... And you would end, finally, at the same point: relying on the Jordan canonical form for actual computations. So, the Lagrange-Sylvester device allows you to get rid of convergence issues -if you're willing to accept $f(A) = p(A)$ as a sound definition.

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Thanks for the answer! Most of the stuff in this answer is new to me. I was thinking about this, and we get $(fg)(A) = f(A)g(A)$ using either the polynomial definition or the power series definition. Thus if $f(t) = 1/t$ and $g(t) = tf(t)$, we get $A \cdot f(A) = g(A) = I$, which is why finding the inverse this way works, and similarly certain properties of $\sqrt{x}$ and other functions are retained. Or in other words, we get the inverses and roots this way because the the map $f \mapsto f(A)$ is a ring homomorphism. Does this make sense? –  Mikko Korhonen Sep 1 '12 at 22:44
    
The thing that was puzzling me was why for functions $f(x)$ the corresponding matrix functions $f(A)$ have similar properties, and also why this all works out with the power series definition. That last part in your answer cleared this up for me. –  Mikko Korhonen Sep 1 '12 at 22:50
    
Think you meant $g(t) = t$, didn't you? On the other hand, besides the last theorem, there's a subtle point: in order to define $f(A)$ by means of its Taylor series (namely, to be an analytic function), you need something stronger than to define by means of the Lagrange-Sylvester (just to be defined on the spectrum of $A$). So, though when you have both definitions available, they agree, this second one is available for a larger class of functions and matrices. –  a.r. Sep 2 '12 at 2:29

When the matrix $A$ is not diagonalizable, working with the "spectrum" becomes technically involved, but even then via Cauchy's theorem of complex analysis you can make sense of $\sin(A)$, etc., in a single formula not involving any eigenvalues or diagonalization process: When all eigenvalues of $A$ are contained in the disk $D_R$ of radius $R$ around $0$ then $$\sin(A)={1\over2\pi i}\int_{\partial D_R}{\sin z\over z-A}\ dz\ .$$

Back to matrices: In the one-dimensional case an $A:\ {\mathbb R}\to{\mathbb R}$ is just a scaling $t\mapsto \lambda t$ with a certain factor $\lambda\in{\mathbb C}$, and the only meaning $f(A)$ could have then is that $f(A)$ scales by the factor $f(\lambda)$.

In the $n$-dimensional diagonalizable case the space $V={\mathbb R}^n$ where $A$ acts can be written as a direct sum $V=\oplus_{i=1}^n V_i$ of the one-dimensional eigenspaces $V_i:=\{x| Ax=\lambda_i x\}$. This means that $A$ acts as $n$ independent one-dimensional scalings $t\mapsto \lambda_i t$ in parallel. If you are given a function $f$ whose domain $\Omega\subset{\mathbb C}$ contains the $\lambda_i$, then $f(A)$ restricted to one of the $V_i$ is just the map $$f(A)\restriction V_i: \quad t\mapsto f(\lambda_i) t\ ,$$ where $t$ is a "local" coordinate in $V_i$, denoted by $t_i$ when we look at all of these at the same time.

Now, since all this makes sense when we look at $A$ as being a linear map $V\to V$, by general principles this also makes sense in the world of matrices. In particular: If $T$ is the matrix whose columns are the eigenvectors of $A$, i.e., the basis vectors of the $V_i$ above, and $D={\rm diag}(\lambda_1,\ldots,\lambda_n)$ is the diagonal matrix containing the eigenvalues, then $$f(A)=T\ {\rm diag}\bigl(f(\lambda_1),\ldots,f(\lambda_n)\bigr)\ T^{-1}\ .$$

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