\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}

\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}

\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}

\def\braket#1#2{\langle #1|#2\rangle}

\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}

\def\e{\mathcal{E}}

\def\tr{\mathrm{tr}}

\)

**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.

\]

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