Normal matrix

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In mathematics, a complex square matrix Template:Mvar is normal if

where A is the conjugate transpose of Template:Mvar. That is, a matrix is normal if it commutes with its conjugate transpose.

A real square matrix Template:Mvar satisfies A = AT, and is therefore normal if ATA = AAT.

Normality is a convenient test for diagonalizability: a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix Template:Mvar satisfying the equation AA = AA is diagonalizable.

The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

Special cases

Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal. However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian. For example

is neither unitary, Hermitian, nor skew-Hermitian, yet it is normal because

Consequences

Proposition. A normal triangular matrix is diagonal.

Let Template:Mvar be a normal upper triangular matrix. Since (AA)ii = (AA)ii, the first row must have the same norm as the first column:

The first entry of row 1 and column 1 are the same, and the rest of column 1 is zero. This implies the first row must be zero for entries 2 through Template:Mvar. Continuing this argument for row–column pairs 2 through Template:Mvar shows Template:Mvar is diagonal.

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies:

Proposition. A matrix Template:Mvar is normal if and only if there exists a diagonal matrix Λ and a unitary matrix Template:Mvar such that A = UΛU.

The diagonal entries of Λ are the eigenvalues of Template:Mvar, and the columns of Template:Mvar are the eigenvectors of Template:Mvar. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of Template:Mvar.

Another way of stating the spectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of Cn. Phrased differently: a matrix is normal if and only if its eigenspaces span Cn and are pairwise orthogonal with respect to the standard inner product of Cn.

The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let Template:Mvar be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, Template:Mvar. If Template:Mvar is normal, so is Template:Mvar. But then Template:Mvar must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal.

The spectral theorem permits the classification of normal matrices in terms of their spectra, for example:

Proposition. A normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane.
Proposition. A normal matrix is self-adjoint if and only if its spectrum is contained in R.

In general, the sum or product of two normal matrices need not be normal. However, the following holds:

Proposition. If Template:Mvar and Template:Mvar are normal with AB = BA, then both AB and A + B are also normal. Furthermore there exists a unitary matrix Template:Mvar such that UAU and UBU are diagonal matrices. In other words Template:Mvar and Template:Mvar are simultaneously diagonalizable.

In this special case, the columns of U are eigenvectors of both Template:Mvar and Template:Mvar and form an orthonormal basis in Cn. This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.

Equivalent definitions

It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let Template:Mvar be a n × n complex matrix. Then the following are equivalent:

  1. Template:Mvar is normal.
  2. Template:Mvar is diagonalizable by a unitary matrix.
  3. The entire space is spanned by some orthonormal set of eigenvectors of Template:Mvar.
  4. Template:!!AxTemplate:!! = Template:!!AxTemplate:!! for every Template:Mvar.
  5. The Frobenius norm of Template:Mvar can be computed by the eigenvalues of Template:Mvar:
  6. The Hermitian part {{ safesubst:#invoke:Unsubst||$B=1/2}}(A + A) and skew-Hermitian part {{ safesubst:#invoke:Unsubst||$B=1/2}}(AA) of Template:Mvar commute.
  7. A is a polynomial (of degree n − 1) in Template:Mvar.[1]
  8. A = AU for some unitary matrix Template:Mvar.[2]
  9. Template:Mvar and Template:Mvar commute, where we have the polar decomposition A = UP with a unitary matrix Template:Mvar and some positive semidefinite matrix Template:Mvar.
  10. Template:Mvar commutes with some normal matrix Template:Mvar with distinct eigenvalues.
  11. σi = |λi| for all 1 ≤ in where Template:Mvar has singular values σ1 ≥ ... ≥ σn and eigenvalues |λ1| ≥ ... ≥ |λn|.[3]
  12. The operator norm of a normal matrix Template:Mvar equals the numerical and spectral radii of Template:Mvar. (This fact generalizes to normal operators.) Explicitly, this means:

Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is only quasinormal.

Analogy

It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:

As a special case, the complex numbers may be embedded in the normal 2 × 2 real matrices by the mapping

which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.

Notes

  1. Proof: When Template:Mvar is normal, use Lagrange's interpolation formula to construct a polynomial Template:Mvar such that Template:Overline = P(λj), where λj are the eigenvalues of Template:Mvar.
  2. Horn, pp. 109
  3. {{#invoke:citation/CS1|citation |CitationClass=book }}

References

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