# 2.7 functions of operators

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1.
A linear operator $$A:U\to U$$ that maps a space into itself is called endomorphism. Such operators can be composed with impunity. In elevated language, they form an algebra. (It is a generalization of our representation of complex numbers as $$2\times 2$$-matrices).

2.
Which means we can form some functions of operators. Polynomials are the easiest one: if $$P=a_0x^n+\ldots+a_n$$, then
$P(A)=a_0A^n+\ldots+a_n E.$

Other functions can be defined as well, if they can be approximated by polynomials: the exponential is a familiar example:
$\exp(A)=\sum_{k\geq 0} A^k/k!$
Of course, one needs to make sure that this expression makes sense, that is converge (it does – can be proved using the notion of a matrix norm). So one can form functions like $$\sin(A)$$. These functions can be very useful, but it is hard to beat the matrix exponential.

3.
Indeed, just like the usual exponential function $$\e(t)=\exp(at)$$ satisfies the differential equation
$$d\e/dt=a\e$$ with initial condition $$\e(0)=1$$, the matrix exponential $$\e(t)=\exp(At)$$ satisfies
$\frac{\e(t)}{dt}=A\e(t), \e(o)=E.$

One example of matrix exponential we know already:
$\exp\left(\begin{array}{cc}0&1\\-1&0\end{array}\right)t$
is the matrix of rotation by $$t$$. This survives to dimension 3: take any skew symmetric matrix
$\left(\begin{array}{ccc} 0&c&-b\\ -c&0&a\\ b&-a&0 \end{array} \right)$
then its matrix exponential is a rotation around the axis $$(a,b,c)$$…

Compositions (matrix products of rotations) are rotations again. So the rotations around various axes form a group (of rotations), called SO(3). We’ll encounter it (and more generally, discuss what a group is) later on.

4.
Unlike products of numbers (elements of a field, like reals or rationals), the product of matrices is in general noncommutative: $$AB$$ is not always the same as $$BA$$. The difference is called the commutator:
$[A,B]=AB-BA.$
This leads to some complications: say,
$$\exp(A+B)$$ is not always the same as $$\exp(A)\exp(B)$$ (but it is if $$AB=BA$$).

This noncommutativity is not always bad: the noncommutativity of rotation operators in 3D is responsible for big chunk of our physics.

Exercise: Find the product of rotations by $$\pi/2$$ around $$x$$, $$y$$ and $$z$$ axes.

Exercise: let $$A$$ is the diagonal matrix with elements $$1,2,\ldots, d$$ on the diagonal. Find all matrices commuting with $$A$$.

5.
Quite often knowing that a function of an operator vanishes allows one to characterize the operator.

Example: If $$A^2-A=0$$, then operator is a projector: the space is split $$U=U_0\oplus U_1$$, and
$A|_{U_1}=E_{U_1}, A|_{U_0}=0.$

Another example: nilpotent operators, that is such that a power of it vanishes: $$A$$ is nilpotent if $$A\neq 0, A^n=0$$ for some $$n>1$$. One can show that any nilpotent operator is an upper-triangular matrix in some basis.

6.
If we have an operator in $$L(U,V)$$ and bases in either of the spaces, we obtain the matrix of the operator.
Changing the basis (in $$U$$ or in $$V$$) leads to multiplications of the matrix $$A_{ij}$$ on the right and on the left by the change of the basis matrices:
$A=\sum_{i,j}\ket{f_i}A_{ij}\bra{e_j}=\sum_{i’}\ket{f’_{i’}}\bra{f’_{i’}}\sum_{i,j}\ket{f_i}A_{ij}\bra{e_j}\sum_j\ket{e’_{j’}}\bra{e_{j’}}=\sum_{i’j’}A’_{i’j’}\ket{f’_{i’}}\bra{e_{j’}},$
where the coefficients $$A’_{i’j’}$$ of the matrix of operator $$A$$ in the new bases are given by
$A’_{i’j’}=\sum_{i,j}B_{i’i}A_{ij}C_{jj’}=\sum_{i,j}\braket{e’_{i’}}{e_i}A_{ij}\braket{f_{j}}{f’_{j’}}$
(with the $$B$$ and $$C$$ being the invertible matrices of replacing the basis).
In the case of endomorphisms, i.e. when $$U=V$$, the change of basis matrices are the reciprocal: $$BC=E$$. Alternatively, if $$(A_{ij})$$ is the matrix of the endomorphism $$A$$ in a basis $$e=f$$, then the matrix in the basis $$e’=f’$$ is given by
$A’=BAB^{-1},$
where $$B$$ is the basis change matrix.

One nice property of the functions of operators approximable by polynomials is that to change basis, one need not replace basis before applying the function:
$f(BAB^{-1})=Bf(A)B^{-1}.$

7.
If a $$\Real$$ or $$\Comp$$-valued function of a matrix has the similar property,
$f(BAB^{-1})=f(A),$
such a function is called central.

Immediate examples are $$\det(A)$$ and the coefficients of the characteristic polynomial,
$p_A(z)=\det(zE-A)=z^n-\tr(A)z^{n-1}+\ldots+(-1)^n\det(A).$

8.
In fact all the central functions of a matrix (or operator) are representable as functions of the coefficients of the characteristic polynomial, or as $$\det(f(A))$$ for some operator function of $$A$$, or on the spectrum of $$A$$ (the roots of the characteristic polynomial).

Another characterization of central functions:

$f(AB-BA)=0$

for any $$A,B$$.

Exercise:

Do there exist matrices $$A,B: AB-BA=E$$?

9.

Characteristic polynomial is useful as it allows one to capture eigenvectors, 1-dimensional subspaces left invariant by the operator: we say that $$\lambda$$ is an eigenvalue, and $$v$$ the corresponding eigenvector if
$Av=\lambda v.$

Example: if characteristic polynomial has only simple roots, there exist a basis consisting of eigenvectors.

For a polynomial (or analytic function) $$f$$, and eigenvalue/vector $$\lambda,v$$, one can immediately see that
$f(A)v=f(\lambda)v.$

In particular, if $$\lambda$$ is an element of the spectrum of $$A$$ (i.e., if $$p_A(\lambda)=0$$), then there exists a corresponding eigenvector.

10.

One can prove the following remarkable

Theorem (Cayley-Hamilton):
$p_A(A)=0.$