Lecture 11

[[lecture-data]]

2024-09-20

Readings

2. Chapter 2

Claim: $A \in M_{n}$ is unitary if and only if there exists a $B \in M_{n}$ skew hermitian such that $A = e^{B}$

Skew hermitian means that $B^{*} = - B$ . In particular, $B$ is normal. Lets say that we unitarily diagonalize $B$ so we can write $B = U D U^{*}$ where $U \in M_{n}$ is unitary and $D \in M_{n}$ is diagonal. Then

\begin{aligned} B^{*} & = - B \\ (U D U^{*})^{*} & = - U D U^{*} \\ U D^{*} U^{*} & = - U D U^{*} \\ ⟹ D^{*} & = - D \end{aligned}

And this implies that the elements of $D$ are purely imaginary.

Matrix exponential

The matrix exponential is

e^{B} = \sum_{i = 0}^{\infty} \frac{1}{i!} B^{i}

Theorem

$A \in M_{n}$ is unitary if and only if there exists a $B \in M_{n}$ skew hermitian such that $A = e^{B}$
( see a matrix is unitary if and only if it equals the exponential of a skew hermitian matrix)

Proof

$⟸$ Suppose $B$ is skew hermitian and $A = e^{B}$ . $B$ is normal and we can say $B = U D U^{*}$ where $U$ is unitary and $D$ is diagonal. Recall that the elements of the diagonal of $D$ are pure imaginary. ie, there exist $θ_{1}, \dots, θ_{n} \in R$ such that >$$\begin{bmatrix} \theta_{1} & 0 & \dots & 0 \
0 & \theta_{2} & \ddots & \vdots & \
\vdots & \ddots & \ddots & 0 \
0 & \dots & 0 & \theta_{n}
\end{bmatrix}$$
Then we have

\begin{aligned} A & = e^{B} \\ = e^{U D U^{*}} \\ = U e^{D} U^{*} \\ = U [\begin{array}{c} e^{θ_{1}} & 0 & \dots & 0 \\ 0 & e^{θ_{2}} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & e^{θ_{n}} \end{array}] U^{*} \end{aligned}

where and each of the matrices in the last product are unitary! And so $A$ is unitary, since the product of unitary matrices is unitary

$⟹$ Now, suppose $A$ is unitary. So $A$ is normal, say $A = W Ω W^{*}$ . Then

Ω = [\begin{matrix} ω_{1} & 0 & \dots & 0 \\ 0 & ω 2 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & ω n \end{matrix}]

And since $A$ is unitary, it is an isometry, which implies that each of the eigenvalues of $A$ have modulus 1 ( $| | ω_{i} | | = 1 \forall i$ ). Thus, we can express each $ω_{i}$ as an angle between it and the real axis and this is given by $e^{δ_{i}}$ for $δ_{i} \in R$ . So

Ω = [\begin{matrix} e^{δ_{i}} & 0 & \dots & 0 \\ 0 & e^{δ_{i}} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & e^{δ_{i}} \end{matrix}]

Define

Σ = [\begin{matrix} δ 1 & 0 & \dots & 0 \\ 0 & δ 2 & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & δ n \end{matrix}]

Then $B = W Σ W^{*}$ is skew hermitian ! Then

e^{B} = e^{W Σ W^{*}} = W e^{Σ} W = W Ω W = A

3. Chapter 3 : Jordan Decomposition

Classifying matrices according to equivalence classes of similarity.

Jordan Block

$A$ is a $k \times k$ Jordan block (denoted $J_{k} (λ)$ ) is

[\begin{matrix} λ & 1 & 0 & \dots & 0 \\ 0 & λ & 1 & ⋱ & ⋮ \\ ⋮ & ⋱ & λ & 1 & 0 \\ ⋮ & ⋱ & λ & 1 \\ 0 & \dots & \dots & 0 & λ \end{matrix}]

And this has eigenvalue $λ$ with algebraic multiplicity $k$ . The eigenvectors
$x : [λ I - J_{k} (λ)] x = 0$ can be found to be $x_{1} = anything$ and $x_{2} = \dots = x_{n} = 0$ , so the eigenvector has geometric multiplicity 1.

(see Jordan block)

When the eigenvalue of zero, when we raise it to the order of the block $k$ , we get back the $0$ th matrix. We can see that

[J_{k} (0)^{ℓ}]_{i j} = {\begin{cases} 1 if ℓ = j - i \\ 0 otherwise \end{cases}

and $rank [J_{k} (0)^{ℓ}] = (k - ℓ)_{\geq 0}$

Jordan Matrix

A Jordan Matrix is the direct sum of Jordan blocks.

J = J_{n_{1}} (λ_{1}) \oplus J_{n_{2}} (λ_{2}) \oplus \dots \oplus J_{n_{s}} (λ_{s}) = [\begin{matrix} J_{n_{1}} (λ_{1}) & 0 & \dots & 0 \\ 0 & J_{n_{2}} (λ_{2}) & ⋱ & ⋱ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & J_{n_{s}} (λ_{n}) \end{matrix}]

" $= \oplus_{i = 1}^{s} J_{n_{i}} (λ_{i})$ " and $λ_{i}$ not necessarily distinct.

This is a block diagonal matrix with the jordan blocks along the diagonal.

(see Jordan Matrix)

Theorem

For each $A \in M_{n}$ there exists a Jordan Matrix $J$ such that $A$ is similar to $J$ :

A = S J S^{- 1}

for $S \in M_{n}$ . Further, $J$ is unique up to reordering its Jordan blocks.

Example

$A \in M_{3}$ with all eigenvalues $π$ has 3 possible similarity classes, one for each size of largest Jordan blocks.

(see Jordan's Theorem)

Note

A matrix is diagonalizable if and only if each of its jordan blocks are of size 1.