Matrices, Eigenvalues, and the Size of a Linear Transformation

Christopher Hammond Connecticut College

May 20, 2004

Definition. A linear transformation T : ℝ^{n} → ℝ^{m} is a function such that

T (x + y) = T (x) + T (y),
T (αx) = αT (x)

for all x and y in ℝ^{n} and all α in ℝ.

Such a linear transformation can be represented by an m × n matrix.

Definition. The number λ is an eigenvalue for a linear transformation T : ℝ^{n} → ℝ^{n} if there exists some nonzero x in ℝ^{n} (an eigenvector) such that T (x) = λx.

We determine the eigenvalues of T by finding the zeros of the characteristic polynomial

χ(λ) = det (λI_{n} − M).

Definition. The inner product (or dot product) of two vectors x and y in ℝ^{n} is defined by the rule

⟨x, y⟩ = x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3} + … + x_{n}y_{n}.

Definition. The norm of a vector x in ℝ^{n} is defined by the rule

∥x∥ = √⟨x, x⟩
= √x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + … + x_{n}^{2} .

Definition. The norm of a linear transformation T : ℝ^{n} → ℝ^{n} is defined by the rule

∥T ∥ = sup_{x∈ℝ^{n}, x≠0} \frac{∥T (x)∥}{∥x∥} .

If ∥T (x)∥ = ∥T ∥ ∥x∥ for x ≠ 0, then we say that T attains its norm on the vector x.

Examples.

(1) T (x) = 0.

(2) T (x) = x

(3) T (x) = −2x.

(4) T ([\begin{matrix}a \\ b \end{matrix}]) = [\begin{matrix}−3b \\ 3a \end{matrix}]

(5) T ([\begin{matrix}a \\ b \end{matrix}]) = [\begin{matrix}2b \\ b \end{matrix} ]

(6) T ([\begin{matrix}a \\ b \end{matrix}]) = [\begin{matrix}0 \\ 2a + b \end{matrix} ]

(7) T ([\begin{matrix}a \\ b \end{matrix}]) = [\begin{matrix}a + 2b \\ 3a + 4b \end{matrix} ]

Let T : ℝ^{n} → ℝ^{n} be a linear transformation.

Definition. The adjoint T^{∗} of T is the linear transformation with the property that

⟨T (x), y⟩ = ⟨x, T^{∗}(y)⟩

for all x and y in ℝ^{n}.

Facts:

(1) If T is represented by the matrix M, then T^{∗} is represented by the transpose of M.

(2) ∥T^{∗}∥ = ∥T ∥.

Consider the linear transformation T^{∗}T. One can show that ∥T ∥^{2} is the largest eigenvalue of T^{∗}T; in fact, we have the following result.

Proposition. Let x be a vector in ℝ^{n}; then ∥T (x)∥ = ∥T ∥ ∥x∥ if and only if (T^{∗}T)(x) = ∥T ∥^{2} x.

In other words, T attains its norm on x if and only if x is an eigenvector for T^{∗}T corresponding to the eigenvalue ∥T ∥^{2}.

Let H^{2} denote the set of all analytic functions f(z) = ∑ a_{n}z^{n} on 𝔻 = {z ∈ ℂ : |z| < 1} such that

∥f∥ = ; ; ; √ ∞ ; ∑ ; n=0 |a_{n}|^{2} < ∞.

We call H^{2} the Hardy space.

We define the inner product of two functions f(z) = ∑ a_{n}z^{n} and g(z) = ∑ b_{n}z^{n} in H^{2} by the rule

⟨f, g⟩ = ∞ ; ∑ ; n=0 a_{n}\overline{b_{n}}.

Let φ be an analytic map that takes 𝔻 into 𝔻. We define the composition operator C_{φ} : H^{2} → H^{2} by the rule

C_{φ}(f) = f ∘ φ.

(Operator means linear transformation.) We would like to calculate

∥C_{φ}∥ = sup_{f∈H^{2},f≠0} \frac{∥C_{φ}(f)∥}{∥f∥} .

Question: How do we do this?

Answer: Find the largest eigenvalue of C_{φ}^{∗}C_{φ}.

Cowen’s Adjoint Formula [1988]. Suppose that φ : 𝔻 → 𝔻 is a linear fractional map; that is,

φ(z) = \frac{az + b}{cz + d} .

Then C_{φ}^{∗}(f) is given by the formula

γ(z) (\overline{c} (\frac{f(σ(z)) − f(0)}{σ(z)} ) + \overline{d}f(σ(z))) ,

where

γ(z) = \frac{1}{−\overline{b}z + \overline{d}}

and

σ(z) = \frac{\overline{a}z − \overline{c}}{−\overline{b}z + \overline{d}} .

Theorem [Cowen 1988]. Take φ(z) = az + b with |a| + |b| ≤ 1. Then ∥C_{φ}∥ equals

 ;  ;  ;  ; √\frac{2}{1 + |a|^{2} − |b|^{2} + √(1 − |a|^{2} + |b|^{2}) ^{2} − 4 |b|^{2}}

(Moreover, one can identify the elements on which C_{φ} attains its norm.)

Theorem [H 2003]. Let n be a positive integer and r a real number greater than n. Take

φ(z) = \frac{rz − n}{−(n + 1)z + (r + 1)} .

In this case, ∥C_{φ}∥^{2} is the largest zero of the polynomial

p(λ) = λ^{n+1} − n ; ∑ ; k=0 \begin{matrix}(n + 1) & (r + k + 1) \\ (k + 1) & (r + n + 1) \end{matrix}

· [ k−1 ; ∏ ; m=0 \begin{matrix}(r − n) & (m − n) & (r + m + 1) \\ (m + 1) & (r + n + 1) & (r + m − n + 1) \end{matrix} ] λ^{n−k}.

For example, take

φ(z) = \frac{4z − 2}{−3z + 5} .

We can solve a cubic equation to see that

∥C_{φ}∥^{2} = \frac{5}{7} + \frac{2 Re \sqrt[3]{10 + 5i}}{7}
= \frac{5}{7} + \frac{2√5}{7} cos (\frac{arctan (\frac{1}{2})}{3})
≈1.345547525.

Theorem [Carswell, MacCluer, Schuster 2003]. Let φ(z) = az + b, with |a| ≤ 1 and b = 0 if |a| = 1. Then the norm of C_{φ} acting on the Fock space F is given by the formula

∥C_{φ}∥ = exp (\frac{|b|^{2}}{4 (1 − |a|^{2})} ).