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

Proof. Since \((U x_{i}, U x_{j})=(x_{i}, x_{j})=\delta_{i j}\) , it follows that \(U \mathcal{X}\) is an orthonormal set along with \(\mathcal{X}\) ; it is complete if \(\mathcal{X}\) is, since \((x, U x_{i})=0\) for \(i=1, \ldots, n\) implies that \((U^{*} x, x_{i})=0\) and hence that \(U^{*} x=x=0\) . If, conversely, \(U\mathcal{X}\) is a complete orthonormal set along with \(\mathcal{X}\) , then we have \((U x, U y)=(x, y)\) whenever \(x\) and \(y\) are in \(\mathcal{X}\) , 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 \[\sum_{k} \bar{u}_{k i} u_{k j}=\delta_{i j},\] 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 \(\mathcal{V}\) , then there exists an orthonormal basis \(\mathcal{X}\) in \(\mathcal{V}\) such that the matrix \([A; \mathcal{X}]\) 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 \(\mathcal{X}=\{x_{1}, \ldots, x_{n}\}\) with the property that \(x_{1},\ldots, x_{j}\) lie in \(\mathcal{M}_{j}\) and span \(\mathcal{M}_{j}\) for \(j=1, \ldots, n\) , 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 \(\mathcal{X}\) 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 \(x_{1}, \ldots, x_{j}\) and lies therefore in \(\mathcal{M}_{j}\) . ◻

EXERCISES

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

Exercise 2. For which values of \(\alpha\) are the following matrices isometric?

2 

  1. \(\begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}\) .
  2. \(\begin{bmatrix} \alpha & \frac{1}{2}\\ -\frac{1}{2} & \alpha \end{bmatrix}\) .

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 \((x_{1}, \ldots, x_{k})\) and \((y_{1}, \ldots, y_{k})\) 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 \(U x_{i}=y_{i}\) , \(i=1, \ldots, k\) , is that \((x_{1}, \ldots, x_{k})\) and \((y_{1}, \ldots, y_{k})\) have the same Gramian.

Exercise 7. The mapping \(\xi \to \frac{\xi+1}{\xi-1}\) 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.

  1. If \(A\) is skew, then \(A-1\) is invertible.
  2. If \(U=(A+1)(A-1)^{-1}\) , then \(U\) is isometric. (Hint: \(\|(A+1) y\|^{2}=\|(A-1) y\|^{2}\) for every \(y\) .)
  3. \(U-1\) is invertible.
  4. 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 \(\mathcal{V}\) onto itself (that is, if \(x\) is in \(\mathcal{V}\) , then \(U x\) is in \(\mathcal{V}\) , and if \(y\) is in \(\mathcal{V}\) , then \(y=U x\) for some \(x\) in \(\mathcal{V}\) ), in such a way that \((U x, U y)=(x, y)\) for all \(x\) and \(y\) .

  1. 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 \(y\) .
  2. 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 \(y\) .

  1. Give an example of a conjugation.
  2. Prove that \((J x, y)=(J y, x)\) .
  3. Prove that \(J(x+y)=J x+J y\) .
  4. Prove that \(J(\alpha x)=\bar{\alpha} \cdot J x\) .

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

  1. Give an example of a Hermitian transformation that is not real, and give an example of a real transformation that is not Hermitian.
  2. If \(A\) is real, then the spectrum of \(A\) is symmetric about the real axis.
  3. 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 \(\operatorname{tr} A=0\) , 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.)