# Normal matrix

In mathematics, a complex square matrix *A* is **normal** if

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**=*A*^{T}, and is therefore normal if *A*^{T}*A* = *AA*^{T}.

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

is normal because

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*

^{*}

*)*, the first row must have the same norm as the first column,

_{ii}- .

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

where

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 **C**^{n}. Phrased differently: a matrix is normal if and only if its eigenspaces span **C**^{n} and are pairwise orthogonal with respect to the standard inner product of **C**^{n}.

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

**C**

^{n}. 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:

*A*is normal.*A*is diagonalizable by a unitary matrix.- The entire space is spanned by some orthonormal set of eigenvectors of
*A*. - for every
*x*. - (That is, the Frobenius norm of
*A*can be computed by the eigenvalues of*A*.) - The Hermitian part and skew-Hermitian part of
*A*commute. - is a polynomial (of degree ≤ n − 1) in .
^{1} - for some unitary matrix
*U*.^{2} *U*and*P*commute, where we have the polar decomposition*A*=*UP*with a unitary matrix*U*and some positive semidefinite matrix*P*.*A*commutes with some normal matrix*N*with distinct eigenvalues.- for all where
*A*has singular values and eigenvalues^{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:

## 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:

- Invertible matrices are analogous to non-zero complex numbers
- The conjugate transpose is analogous to the complex conjugate
- Unitary matrices are analogous to complex numbers whose absolute value is 1
- Hermitian matrices are analogous to real numbers
- Hermitian positive definite matrices are analogous to positive real numbers
- Skew Hermitian matrices are analogous to purely imaginary numbers

(As a special case, the complex numbers may be embedded in the normal 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

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

## References

- Horn, Roger A.; Johnson, Charles R. (1985),
*Matrix Analysis*, Cambridge University Press, ISBN 978-0-521-38632-6.

HPTS - Area Progetti - Edu-Soft - JavaEdu - N.Saperi - Ass.Scuola.. - TS BCTV - TS VideoRes - TSODP - TRTWE | ||