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Normal, Self-Adjoint and Unitary Operators & Matrices

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A note on normal, self-adjoint and unitary operators and matrices.

This note is heavily based on the textbook "Linear Algebra" by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence.

Definitions

Let be either or .

Definition 1:

  • A linear operator over a inner product space is normal if and only if .
  • A matrix is normal if and only if .

Definition 2:

  • A linear operator over a inner product space is self-adjoint (Hermitian) if and only if .
  • A matrix is self-adjoint (Hermitian) if and only if .

Definition 3:

  • A linear operator over a complex inner product space is unitary if and only if .
  • A matrix is unitary if and only if .

Definition 4:

  • A linear operator over a real inner product space is orthogonal if and only if .
  • A matrix is orthogonal if and only if .

Evidently, all unitary, orthogonal and self-adjoint linear operators are normal.

Theorems

Schur's Theorem for Linear Operators

Theorem 6.14(Schur): Let be a linear operator on a finite-dimensional inner product space . If the characteristic polynomial of splits, then there exists an orthonormal basis for such that is upper triangular.

To prove this theorem, a lemma is needed.

Lemma: Let be a linear operator on a finite-dimensional inner product space . If has at least one eigenvector, then also has at least one eigenvector.

Proof: Suppose for some and . Then we have

Thus

We can see that is perpendicular to the range of .

Thus the linear operator is not invertible, and it follows that the null space is nontrivial. This means that has at least one eigenvector corresponding to the eigenvalue .

Proof of Schur's Theorem: By mathematical induction on , all we need to do is to find such that is -invariant. i.e.

which is equivalent to saying

According to the lemma, we can find an eigenvector of , and the proof is complete.

Properties of Normal Operators

Theorem 6.15: Let be an inner product space, and be a normal operator on . Then

    1. for all .
    1. is normal for all .
    1. If is an eigenvector of corresponding to the eigenvalue , then is also an eigenvector of corresponding to the eigenvalue .
    1. If and are distinct eigenvalues of corresponding to eigenvectors and , then and are orthogonal.

A rigorous proof can be found in the textbook. This note will focus on giving intuitions and corollaries for these properties rather than proving them one by one.

    1. When is normal, the norm of is the same as the norm of for all . An immediate consequence is that . i.e. The null space of is the same as the null space of . Note that this corollary is nontrivial, since it does not hold for arbitrary linear operators.
    1. Adding a multiple of the identity operator to a normal operator does not affect its normality. As a remark, when adding a mutliple of the identity operator to a self-adjoint operator, a unitary operator or an orthogonal operator, the property of self-adjointness, unitarity or orthogonality may not be preserved, respectively.
    1. For any linear operator , when is an eigenvalue of , is an eigenvalue of (by Lemma). However, the eigenspaces may not be the same. With the normality assumption, the eigenspaces are always the same.
    1. Eigenspaces corresponding to distinct eigenvalues of a normal operator are orthogonal. This property is crucial for constructing an orthonormal basis of eigenvectors for a normal operator(which we will discuss right after this theorem).

Orthonormal basis of Eigenvectors on a Complex Inner Product Space

Theorem 6.16: Let be a linear operator on a finite-dimensional complex inner product space . Then is normal if and only if there exists an orthonormal basis for consisting of eigenvectors of .

Proof: If such an orthonormal basis consisting of eigenvectors of exists, then is normal because diagonal matrices are normal. Now we prove the converse.

By Schur's theorem, we can find an orthonormal basis for such that is upper triangular.

We claim that is diagonal. Suppose the upper-left submatrix of is diagonal. Then we have

Since the basis is orthonormal, for any , we have

Thus the upper-left submatrix of is diagonal.

By mathematical induction, we can see that is diagonal, and the proof is complete.

Note 1: The theorem does not hold for real inner product spaces, because the characteristic polynomial is not guaranteed to split over , and the condition of Schur's theorem is not satisfied. For instance, let be rotation by , then can be represented by the matrix

Although is normal, it does not have any eigenvalue or eigenvector over .

Note 2: This theorem does not extend to infinite-dimensional inner product spaces.

Orthonormal basis of Eigenvectors on a Real Inner Product Space

Theorem 6.17: Let be a linear operator on a finite-dimensional real inner product space . Then is self-adjoint if and only if there exists an orthonormal basis for consisting of eigenvectors of .

To prove this theorem, all we need to do is to show that splits over .

Lemma: Let be a self-adjoint operator on a finite-dimensional inner product space . Then

    1. Every eigenvalue of is real.
    1. The characteristic polynomial of splits.

The proof of this lemma is omitted here. Please refer to the textbook.

Note: Theorem 6.16 and Theorem 6.17 are similar. The main difference is that Theorem 6.16 applies to complex inner product spaces, while Theorem 6.17 applies to real inner product spaces.

In conclusion, though normality is sufficient to guarantee the existence of an orthonormal basis of eigenvectors for a complex inner product space, self-adjointness, a stronger condition, is required for a real inner product space. The fundamental reason is that is algebraically closed but is not, and polynomials over may not split.

Properties of Unitary and Orthogonal Operators

Theorem 6.18: The following statements are equivalent.

    1. for any .
    1. For any orthonormal basis for , is also an orthonormal basis for .
    1. There exists an orthonormal basis for such that is also an orthonormal basis for .
    1. for all .

Intuitions:

    1. This is the definition of unitary(or orthogonal) operators.
    1. Unitary operators preserve inner products.
  • (c)(d) Orthonormality is preserved under unitary operators.
    1. Unitary operators preserve norms.

It is noteworthy that normality does not imply the preservation of inner products, but unitarity(orthogonality) does.

Several corollaries can be derived from Theorem 6.16, Theorem 6.17 and Theorem 6.18.

Corollary 1: Let be a linear operator on a finite-dimensional real inner product space . Then has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value 1 if and only if is both self-adjoint and orthogonal.

Corollary 2: Let be a linear operator on a finite-dimensional complex inner product space . Then has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value 1 if and only if is unitary.

Matrix Version of Theorems

We can restate Theorem 6.16, Theorem 6.17 and Theorem 6.14 in terms of matrices.

Theorem 6.19: Let . Then is normal if and only if is unitarily equivalent to a diagonal matrix.

Theorem 6.20: Let . Then is symmetric if and only if is orthogonally equivalent to a real diagonal matrix.

Theorem 6.21(Schur): Let be a matrix whose characteristic polynomial splits over . Then

    1. If , then is unitarily equivalent to a complex upper triangular matrix.
    1. If , then is orthogonally equivalent to a real upper triangular matrix.