Normal matrix

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In mathematics, a complex square matrix A is normal if

A^*A=AA^* \

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

A matrix A with real entries 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 A satisfying the equation A*A=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. As an example, the matrix

A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}

is normal because

AA^* = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix} = A^*A.

The matrix A is neither unitary, Hermitian, nor skew-Hermitian.

The sum or product of two normal matrices is not necessarily normal. If they commute, however, then this is true.

If A is both a triangular matrix and a normal matrix, then A is diagonal. This can be seen by looking at the diagonal entries of A*A and AA*, where A is a normal, triangular matrix. Say A is upper-triangular. Because (A*A)ii=(AA*)ii, the first row must have the same norm as the first column,

||A e_1||^2 = ||A^* e_1||^2.

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

Consequences

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies: a matrix A is normal if and only if it can be represented by a diagonal matrix Λ and a unitary matrix U by the formula

 \mathbf{A} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^*

where

 \mathbf{\Lambda} = \operatorname{diag}(\lambda_1, \lambda_2, \dots)
 \mathbf{U}^*\mathbf{U} = \mathbf{U} \mathbf{U}^* = \mathbf{I}.

The entries λ of the diagonal matrix Λ are the eigenvalues of A, and the columns of U are the eigenvectors of A. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of U.

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 can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition. In fact, let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B. But then B 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, a normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane. Also, a normal matrix is self-adjoint if and only if its spectrum consists of reals.

In general, the sum or product of two normal matrices need not be normal. However, there is a special case: if A and B are normal with AB = BA, then both AB and A + B are also normal. Furthermore the two are simultaneously diagonalizable, that is: both A and B are made diagonal by the same unitary matrix U. Both UAU* and UBU* are diagonal matrices. In this special case, the columns of U* are eigenvectors of both A and B 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 A be a n-by-n complex matrix. Then the following are equivalent:

  1. A is normal.
  2. A is diagonalizable by a unitary matrix.
  3. The entire space is spanned by some orthonormal set of eigenvectors of A.
  4. \|Ax\| = \|A^*x\| for every x.
  5. \operatorname{tr} (A^* A) = \sum_j^n |\lambda_j|^2. (That is, the Frobenius norm of A can be computed by the eigenvalues of A.)
  6. The Hermitian part \frac{1}{2} (A + A^*) and skew-Hermitian part \frac{1}{2} (A - A^*) of A commute.
  7. A^* is a polynomial (of degree ≤ n − 1) in A.1
  8. A^* = AU for some unitary matrix U.2
  9. U and P commute, where we have the polar decomposition A = UP with a unitary matrix U and some positive semidefinite matrix P.
  10. A commutes with some normal matrix N with distinct eigenvalues.
  11. \sigma_i(A)=|\lambda_i(A)| for all i=1...n where A has singular values \sigma_1(A)\ge ...\ge\sigma_n(A) and eigenvalues |\lambda_1(A)|\ge ...\ge|\lambda_n(A)|3

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.

The operator norm of a normal matrix N equals the numerical and spectral radii of N. (This fact generalizes to normal operators.) Explicitly, this means:

\sup_{ \|x\|=1 } \|Nx\| =  \sup_{ \|x\|=1 } |\langle Nx, x \rangle| = \max \{ |\lambda| : \lambda \in \sigma(N) \}

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\times 2 real matrices by the mapping a+bi \mapsto \begin{pmatrix}a&b\\-b&a\end{pmatrix}, which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.)

Notes

  1. ^ Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that \overline{\lambda_j} =  P(\lambda_j), where \lambda_j are the eigenvalues of A.
  2. ^ Horn, pp. 109
  3. ^ Horn, Roger A.; Johnson, Charles R. (1991). Topics in Matrix Analysis. Cambridge University Press. p. 157. ISBN 978-0-521-30587-7. 

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