# Proof of Spectral Theorem

There are a number of results called Spectral Theorems. This question deals with the Linear Algebra result on normal operators, which has the self-adjoint case as a particular case.

In class, we saw the spectral theorem for self-adjoint operators, and the teacher attempted a sketch of a proof which nobody in the class understood. Writing my thesis, I hit into the normal operator case, so I was looking for a complete proof of it.

How is it proven?

The main reason for posting this was to answer it, thus collecting all this stuff in a single place for future reference -- and present too.

The first item on this proof is that a linear operator on a finite-dimensional complex vector space admits an upper triangular representation. This is proved by induction on $$n:=\dim V$$, $$V$$ being the vector space. If it is 1D, the proof is trivial. Suppose $$\dim V=n>1$$ and the theorem holds for dimensions up to $$n-1$$. We know our operator $$T$$ has an eigenvalue. Indeed, consider $$v,Tv,T^2v,T^3v,\dotsc,T^nv$$. Those cannot be linearly independent if $$v\neq0$$, since they are $$n+1$$ and $$\dim V=n$$. So there exist $$a_i\in\mathbb{C}$$ such that: $$\sum_{i=1}^nT^iva_i=0.$$ Let $$m$$ be the largest index such that $$a_m\neq0$$. THis is not 0, since $$v\neq0$$. Factor the polynomial: $$a_0+a_1z+\dotso+a_mz^m=c(z-\lambda_1)\cdot\dotso\cdot(z-\lambda_m).$$ Substituting $$T$$ for $$z$$, and applying to $$v$$, we find: $$0=\left(\sum_{i=1}^ma_iT^i\right)v=c(T-\lambda_1I)\cdot\dotso\cdot(T-\lambda_mI)v,$$ so $$T-\lambda_iI$$ is not injective for some $$i$$. But this equates to $$\lambda_i$$ being an eigenvector, since not injective iff has nontrivial kernel iff $$(T-\lambda_iI)v=0$$ for some $$v\neq0$$ iff $$\lambda_iv=Tv$$ i.e. $$\lambda_i$$ is an eigenvector. So going back to our original $$T$$, consider any eigenvalue $$\lambda$$. $$T-\lambda I$$ is not injective, but by nullity+rank we have $$T-\lambda I$$ is not surjective. If $$U=\mathrm{Im}(T-\lambda I)$$ is the range of that operator, then $$\dim U<\dim V$$. Also, $$U$$ is invariant under $$T$$ since: $$Tu=(T-\lambda I)u+\lambda u,$$ and if $$u\in U$$ then both summands are in $$U$$. So $$T|_U$$ is an operator on $$U$$, and by induction there exists a basis of $$U$$ such that $$T$$ is represented by an upper triangular matrix w.r.t that basis. So if $$k:=\dim U$$ and that basis is $$\{u_1,\dotsc,u_k$$, then $$Tu_j$$ is in the span of $$u_1,\dotsc,u_j$$ for all $$j\leq m$$. Extend that basis to a basis of $$V$$ by adding extra vectors $$v_1,\dotsc,v_{n-k}$$. $$Tv_i$$ is in the span of $$u_1,\dotsc,u_k$$ for all $$i\leq n-k$$, thus in that of $$u_1,\dotsc,u_k,v_1,\dotsc,v_i$$. And this gives us upper triangularity of the matrix representing $$T$$ w.r.t. $$u_1,\dotsc,u_k,v_1,\dotsc,v_{n-k}$$, QED.

The rest of this answer is practically copied off this pdf. First of all, notice how $$T$$, a linear operator, is uniquely determined by the values of $$\langle Tu,v\rangle$$ for $$u,v\in V$$. That is because the inner product is positive definite, so if $$S$$ satisfies $$\langle Tu,v\rangle=\langle Su,v\rangle$$ for all $$u,v\in V$$, we first conclude $$\langle(T-S)u,v\rangle=0$$ for all $$u,v\in V$$, but fixing $$u$$ this means $$(T-S)u=0$$, and that holds for all $$u$$, hence $$T-S=0$$ or $$T=S$$. This makes it sensible to define an operator via: $$\langle Tu,v\rangle=\langle u,T^\ast v\rangle,$$ for all $$u,v\in V$$. $$T^\ast$$ is uniquely determined as seen above, and is called the adjoint of $$T$$ w.r.t this inner product. Elementary properties of the operation of taking the adjoint are that $$(S+T)^\ast=S^\ast+T^\ast$$, $$(aS)^\ast=\bar aS^\ast$$ for the complex case, the identity is self-adjoint (i.e. coincides with its adjoint), adjoining is an involution (i.e. $$(T^\ast)^\ast=T$$), $$M(T^\ast)=M(T)^\ast$$ in the complex case, denoting by $$^\ast$$ the conjugate transpose of a matrix, and $$(ST)^\ast=T^\ast S^\ast$$. The linked pdf also proves the eigenvalues of a self-adjoint operators are all real, but this is irrelevant here, so I will leave the proof to that pdf. We define normal operators as those for which $$TT^\ast=T^\ast T$$, i.e. those commuting with their adjoints. The polarization identity is another interesting result I leave to the pdf. One result we will use is that, when $$\|v\|=\sqrt{\langle v,v\rangle}$$, then $$\|Tv\|=\|T^\ast v\|$$ for any $$v$$ if $$T$$ is normal The proof is immediate: \begin{align*} T\text{ is normal}\iff{}&TT^\ast-T^\ast T=0\iff\langle(TT^\ast-T^\ast T)v,v\rangle=0\quad\forall v\in V\iff{} \\ {}\iff{}&\langle T^\ast Tv,v\rangle=\langle TT^\ast v,v\rangle\quad\forall v\in V\iff{} \\ {}\iff{}&\|T^\ast v\|^2=\langle T^\ast v,T^\ast v\rangle=\langle Tv,Tv\rangle=\|Tv\|^2. \end{align*}

As is subsequently proved, this implies that if $$T$$ is normal the kernel of $$T,T^\ast$$ coincide, the eigenvalues of $$T^\ast$$ and $$T$$ are mutually conjugate, and that distinct eigenvalues are associated to orthogonal eigenvectors, which is in fact true in general.

Now the big result: unitary diagonalizability equates to normality. This statement is of course equivalent to proving an operator $$T$$ is normal iff it admits an orthonormal eigenbasis, since any change of basis is unitary. So let us assume $$T$$ is normal. We know any operator can be represented by an upper triangular matrix w.r.t. some basis. We take that basis and show the corresponding matrix representation of $$T$$, $$M(T)$$, is in fact diagonal. This makes use of the Pythagorean theorem, proved here, and of the norm identity we proved a while ago relating the norm of an image via $$T$$ to that via $$T^\ast$$. By definition, if $$M(T)=(a_{ij})_{i,1=1}^n$$, we have $$Te_i=a_{ii}e_i$$,and since $$M(T^\ast)=M(T)^\ast$$ we also know $$T^\ast e_i=\sum_i^n\bar a_{ik}e_k$$. So by the Pythagorean theorem and the norm identity: $$|a_{ii}|^2=\|Te_i\|^2=\|T^\ast e_i\|^2=\sum_{k=i}^n\|a_{ik}|^2,$$ implying those for $$k\neq i$$ are all zero terms. The above holds for any $$i$$, proving $$M(T)$$ is diagonal.

Now suppose $$M(T)$$ is diagonalizable w.r.t. some orthonormal eigenbasis. $$M(T^\ast)=M(T)^\ast$$, so $$T^\ast$$ is also diagonalizable. Indeed, they are both diagonalizable w.r.t. the same basis, since the eigenvalues are mutually conjugate and the eigenvectors coincide. But we know $$M(TT^\ast)=M(T)M(T^\ast)$$, so: $$M(TT^\ast)=M(T)M(T^\ast)=M(T)M(T)^\ast=M(T)^\ast M(T)=M(T^\ast)M(T)=M(T^\ast T),$$ since diagonal matrices always commute. Thus, $$T^\ast T=TT^\ast$$, for if the matrix representations w.r.t. some basis coincide it means the two have the same images for any vector, and thus coincide. So if $$T$$ is diagonalizable, $$T$$ is normal.

Update

I just realised the proof implicitly uses the fact that if the quadratic form associated to an operator is zero then the operator is zero, i.e. $$\langle Tv,v\rangle\,\,\forall v\in V\implies T=0$$. This is proved here on p. 147:

$$\quad$$ (ii) Since $$(T(x+y),x+y)=(Tx,x)+(Tx,y)+(Ty,x)+(Ty,y)$$, $$x,y\in V$$, and $$(Tv,v)=0$$ for all $$v\in V$$, we have $$\tag{*} 0=(Tx,y)+(Ty,x).$$ If $$V$$ is an inner product space over $$\mathbb{R}$$, then from $$(*)$$: $$0=(Tx,y)+(Ty,x)=(Tx,y)+(y,Tx)=2(Tx,y).$$ Hence, $$(Tx,y)=0$$ for all $$x,y\in V$$, and $$T\equiv 0$$.

$$\quad$$ If $$V$$ is an inner product space over $$\Bbb C$$, then replacing $$y$$ by $$iy$$ in $$(*)$$, we have $$(Tx,iy)+(iTy,x)=0$$. Thus for all $$x,y\in V$$: $$(Tx,y)-(Ty,x)=0.$$ Hence, $$(Tx,y)=0$$ for all $$x,y\in V$$, and $$T\equiv 0$$.

• Um, the update is imprecise, that implication holds for self-adjoint operators, as is $TT^\ast-T^\ast T$. – MickG Oct 26 '15 at 15:14
• I can see this is an old post but let me just say that the whole argument only holds for finite dimensional matrices (which may not be entirely clear). – lcv Aug 23 '19 at 11:49

The spectral theorem says that every normal operator$$~\phi$$ on a finite dimensional complex inner product space$$~V$$ is diagonalisable, and that its eigenspaces are mutually orthogonal. As a consequence an orthonormal basis of$$~V$$ consisting eigenvectors for$$~\phi$$ can be chosen.

Here is a simple proof. I will start with the special case where $$\phi$$ is an Hermitian (also called self-adjoint) operator, which is quite easy; more so indeed than the case of a self-adjoint-operator on a finite dimensional real inner product space (also called Euclidean space) that I discussed here. Then I will use this result to generalise it to the case of normal operators.

A basic fact about adjoints is that for any linear operator $$\phi$$ on$$~V$$, whenever a subspace $$W$$ is stable under$$~\phi$$, its orthogonal complement $$W^\perp$$ is stable under its adjoint$$~\phi^*$$. For if $$v\in W^\perp$$ and $$w\in W$$, then $$\langle w\mid \phi^*(v)\rangle=\langle \phi(w)\mid v\rangle=0$$ since $$\phi(w)\in W$$, so that $$\phi^*(v)\in W^\perp$$. Then for a Hermitian operator $$\phi$$ (so with $$\phi^*=\phi$$), the orthogonal complement of any $$\phi$$-stable subspace is again $$\phi$$-stable.

Now we prove the spectral theorem for Hermitian $$\phi$$ by induction on $$\dim V$$. When $$\dim V=0$$ the unique operator $$\phi$$ on $$V$$ is diagonalisable with empty set of eigenvalues, and the result is trivial. Now assuming $$\dim V>0$$, there is at least one eigenvalue (since the characteristic polynomial of $$\phi$$ has a root by the fundamental theorem of algebra) so we can choose an eigenvector $$v_1$$ of$$~\phi$$. The subspace $$W=\langle v_1\rangle$$ it spans is $$\phi$$-stable by the definition of an eigenvector, and so $$W^\perp$$ is $$\phi$$-stable as well. We can then restrict $$\phi$$ to a linear operator on $$W^\perp$$, which is clearly self-adjoint, so our induction hypothesis gives us an orthonormal basis of $$W^\perp$$ consisting of eigenvectors for that restriction; call them $$(v_2,\ldots,v_n)$$. Viewed as elements of $$V$$, the vectors $$v_2,\ldots,v_n$$ are eigenvectors of$$~\phi$$, and clearly the family $$(v_1,\ldots,v_n)$$ is orthonormal. It is an orthonormal basis of eigenvectors of$$~\phi$$, and we are done.

So now I will deduce from this the more general case where $$\phi$$ is a normal operator. First, note that the from Hermitian case easily follows the anti-Hermitian case, i.e., the one where $$\phi^*=-\phi$$. One way is to observe that for anti-Hermitian $$~\phi$$ one still has that if a subspace $$W$$ is $$\phi$$-stable then so is its orthogonal complement $$W^\perp$$ (just put a minus sign in the above argument), so the proof of the spectral theorem for Hermitian$$~\phi$$ can be reused word-for-word. Another way is to observe that $$\phi$$ is Hermitian if and only if $$\def\ii{\mathbf i}\ii\phi$$ is anti-Hermitian, so if one is diagonalisable so is the other, with the same eigenspaces but with the eigenvalues of $$\ii\phi$$ multiplied by$$~\ii$$ with respect to those of$$~\phi$$. (One easily shows the eigenvalues are real in the Hermitian case and imaginary in the anti-Hermitian case, but that is of no importance here.)

Next observe that any linear operator $$\phi$$ can be written as the sum of a Hermitian operator $$\frac12(\phi+\phi^*)$$ and an anti-Hermitian operator $$\frac12(\phi-\phi^*)$$, which are called its Hermitian and anti-Hermitian parts. Moreover $$\phi$$ and $$\phi^*$$ commute (i.e., $$\phi$$ is a normal operator) if and only if the Hermitian and anti-Hermitian parts of $$\phi$$ commute.

So for a normal operator$$~\phi$$ we know that its Hermitian and anti-Hermitian parts are both diagonalisable with mutually orthogonal eigenspaces, and the two parts commute. Now apply the well known fact that two commuting diagonalisable operators can be simultaneously diagonalised; any basis of common eigenvectors is a basis of diagonalisation for$$~\phi$$. Finally, if one has two eigenvectors of$$~\phi$$ for distinct eigenvalues, then they are also eigenvectors for the Hermitian and anti-Hermitian parts of$$~\phi$$, and for at least one of the two they are so for distinct eigenvalues; by virtue of that fact the vectors are orthogonal. This shows that eigenspaces of$$~\phi$$ are mutually orthogonal.

In fact, it remains true for a normal operator$$~\phi$$ that the orthogonal complement of a $$\phi$$-stable subspace is always $$\phi$$-stable, so one could have reused the proof of the Hermitian case once again. However to prove this fact would seem to be more work than the separate argument I gave.