# Normal matrix

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*^{∗} = *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 Template:Mvar 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. 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 (*A*^{∗}*A*)_{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*=*U*Λ*U*^{∗}.

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 **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 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 **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 Template:Mvar be a *n* × *n* complex matrix. Then the following are equivalent:

- Template:Mvar is normal.
- Template:Mvar is diagonalizable by a unitary matrix.
- The entire space is spanned by some orthonormal set of eigenvectors of Template:Mvar.
- Template:!!
*Ax*Template:!! = Template:!!*A*^{∗}*x*Template:!! for every Template:Mvar. - The Frobenius norm of Template:Mvar can be computed by the eigenvalues of Template:Mvar:
- The Hermitian part {{ safesubst:#invoke:Unsubst||$B=1/2}}(
*A*+*A*^{∗}) and skew-Hermitian part {{ safesubst:#invoke:Unsubst||$B=1/2}}(*A*−*A*^{∗}) of Template:Mvar commute. *A*^{∗}is a polynomial (of degree ≤*n*− 1) in Template:Mvar.^{[1]}*A*^{∗}=*AU*for some unitary matrix Template:Mvar.^{[2]}- 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. - Template:Mvar commutes with some normal matrix Template:Mvar with distinct eigenvalues.
*σ*= |_{i}*λ*| for all 1 ≤_{i}*i*≤*n*where Template:Mvar has singular values*σ*_{1}≥ ... ≥*σ*and eigenvalues |_{n}*λ*_{1}| ≥ ... ≥ |*λ*|._{n}^{[3]}- 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:

- 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 with absolute value 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 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

- ↑ Proof: When Template:Mvar is normal, use Lagrange's interpolation formula to construct a polynomial Template:Mvar such that Template:Overline =
*P*(*λ*), where_{j}*λ*are the eigenvalues of Template:Mvar._{j} - ↑ Horn, pp. 109
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}

## References

- {{#invoke:citation/CS1|citation

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