标准正交基的变换

We have seen that the theory of the passage from one linear basis of a vector space to another is best studied by means of an associated linear transformation $A$ (Sections 46, 47); the question arises as to what special properties $A$ has when we pass from one orthonormal basis of an inner product space to another. The answer is easy.

Theorem 1. If $𝒳 = {x_{1}, \dots, x_{n}}$ is an orthonormal basis of an $n$ -dimensional inner product space $𝒱$ , and if $U$ is an isometry on $𝒱$ , then $U 𝒳 = {U x_{1}, \dots, U x_{n}}$ is also an orthonormal basis of $𝒱$ . Conversely, if $U$ is a linear transformation and $𝒳$ is an orthonormal basis with the property that $U 𝒳$ is also an orthonormal basis, then $U$ is an isometry.

Proof. Since $(U x_{i}, U x_{j}) = (x_{i}, x_{j}) = δ_{i j}$ , it follows that $U 𝒳$ is an orthonormal set along with $𝒳$ ; it is complete if $𝒳$ is, since $(x, U x_{i}) = 0$ for $i = 1, \dots, n$ implies that $(U^{*} x, x_{i}) = 0$ and hence that $U^{*} x = x = 0$ . If, conversely, $U 𝒳$ is a complete orthonormal set along with $𝒳$ , then we have $(U x, U y) = (x, y)$ whenever $x$ and $y$ are in $𝒳$ , and it is clear that by linearity we obtain $(U x, U y) = (x, y)$ for all $x$ and $y$ . ◻

We observe that the matrix $(u_{i j})$ of an isometric transformation, with respect to an arbitrary orthonormal basis, satisfies the conditions $\sum_{k} {\bar{u}}_{k i} u_{k j} = δ_{i j},$ and that, conversely, any such matrix, together with an orthonormal basis, defines an isometry. (Proof: $U^{*} U = 1$ . In the real case the bars may be omitted.) For brevity we shall say that a matrix satisfying these conditions is an isometric matrix .

An interesting and easy consequence of our considerations concerning isometries is the following corollary of Section: Triangular form , Theorem 1.

Theorem 2. If $A$ is a linear transformation on a complex $n$ -dimensional inner product space $𝒱$ , then there exists an orthonormal basis $𝒳$ in $𝒱$ such that the matrix $[A; 𝒳]$ is triangular, or equivalently, if $[A]$ is a matrix, then there exists an isometric matrix $[U]$ such that $[U]^{- 1} [A] [U]$ is triangular.

Proof. In Section: Triangular form , in the derivation of Theorem 2 from Theorem 1, we constructed a (linear) basis $𝒳 = {x_{1}, \dots, x_{n}}$ with the property that $x_{1}, \dots, x_{j}$ lie in $ℳ_{j}$ and span $ℳ_{j}$ for $j = 1, \dots, n$ , and we showed that with respect to this basis the matrix of $A$ is triangular. If we knew that this basis is also an orthonormal basis, we could apply Theorem 1 of the present section to obtain the desired result. If $𝒳$ is not an orthonormal basis, it is easy to make it into one; this is precisely what the Gram-Schmidt orthogonalization process ( Section: Complete orthonormal sets ) can do. Here we use a special property of the Gram-Schmidt process, namely, that the $j$ -th element of the orthonormal basis it constructs is a linear combination of $x_{1}, \dots, x_{j}$ and lies therefore in $ℳ_{j}$ . ◻

EXERCISES

Exercise 1. If $(A x) (t) = x (- t)$ on $𝒫$ (with the inner product given by $(x, y) = \int_{0}^{1} x (t) \overset{―}{y (t)} d t$ ) is the linear transformation $A$ isometric? Is it self-adjoint?

Exercise 2. For which values of $α$ are the following matrices isometric?

$[\begin{matrix} α & 0 \\ 1 & 1 \end{matrix}]$ .
$[\begin{matrix} α & \frac{1}{2} \\ - \frac{1}{2} & α \end{matrix}]$ .

Exercise 3. Find a $3$ -by- $3$ isometric matrix whose first row is a multiple of $(1, 1, 1)$ .

Exercise 4. If a linear transformation has any two of the properties of being self-adjoint, isometric, or involutory, then it has the third. (Recall that an involution is a linear transformation $A$ such that $A^{2} = 1$ .)

Exercise 5. If an isometric matrix is triangular, then it is diagonal.

Exercise 6. If $(x_{1}, \dots, x_{k})$ and $(y_{1}, \dots, y_{k})$ are two sequences of vectors in the same inner product space, then a necessary and sufficient condition that there exist an isometry $U$ such that $U x_{i} = y_{i}$ , $i = 1, \dots, k$ , is that $(x_{1}, \dots, x_{k})$ and $(y_{1}, \dots, y_{k})$ have the same Gramian.

Exercise 7. The mapping $ξ \to \frac{ξ + 1}{ξ - 1}$ maps the imaginary axis in the complex plane once around the unit circle, missing the point $1$ ; the inverse mapping (from the circle minus a point to the imaginary axis) is given by the same formula. The transformation analogues of these geometric facts are as follows.

If $A$ is skew, then $A - 1$ is invertible.
If $U = (A + 1) (A - 1)^{- 1}$ , then $U$ is isometric. (Hint: $‖ (A + 1) y ‖^{2} = ‖ (A - 1) y ‖^{2}$ for every $y$ .)
$U - 1$ is invertible.
If $U$ is isometric and $U - 1$ is invertible, and if $A = (U + 1) (U - 1)^{- 1}$ , then $A$ is skew.

Each of $A$ and $U$ is known as the Cayley transform of the other.

Exercise 8. Suppose that $U$ is a transformation (not assumed to be linear) that maps an inner product space $𝒱$ onto itself (that is, if $x$ is in $𝒱$ , then $U x$ is in $𝒱$ , and if $y$ is in $𝒱$ , then $y = U x$ for some $x$ in $𝒱$ ), in such a way that $(U x, U y) = (x, y)$ for all $x$ and $y$ .

Prove that $U$ is one-to-one and that if the inverse transformation is denoted by $U^{- 1}$ , then $(U^{- 1} x, U^{- 1} y) = (x, y)$ and $(U x, y) = (x, U^{- 1} y)$ for all $x$ and $y$ .
Prove that $U$ is linear. (Hint: $(x, U^{- 1} y)$ depends linearly on $x$ .)

Exercise 9. A conjugation is a transformation $J$ (not assumed to be linear) that maps a unitary space onto itself and is such that $J^{2} = 1$ and $(J x, J y) = (y, x)$ for all $x$ and $y$ .

Give an example of a conjugation.
Prove that $(J x, y) = (J y, x)$ .
Prove that $J (x + y) = J x + J y$ .
Prove that $J (α x) = \bar{α} \cdot J x$ .

Exercise 10. A linear transformation $A$ is said to be real with respect to a conjugation $J$ if $A J = J A$ .

Give an example of a Hermitian transformation that is not real, and give an example of a real transformation that is not Hermitian.
If $A$ is real, then the spectrum of $A$ is symmetric about the real axis.
If $A$ is real, then so is $A^{*}$ .

Exercise 11. Section: Change of orthonormal basis , Theorem 2 shows that the triangular form can be achieved by an orthonormal basis; is the same thing true for the Jordan form?

Exercise 12. If $tr A = 0$ , then there exists an isometric matrix $U$ such that all the diagonal entries of $[U]^{- 1} [A] [U]$ are zero. (Hint: see Section: Triangular form , Ex. 6.)