Normal matrix
Template:Short description In mathematics, a complex square matrix Template:Mvar is normal if it commutes with its conjugate transpose Template:Math:
- <math>A \text{ normal} \iff A^*A = AA^* .</math>
The concept of normal matrices can be extended to normal operators on infinite-dimensional normed 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.
The spectral theorem states that 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 Template:Math is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus <math>A = U D U^*</math> and <math>A^* = U D^* U^*</math>where <math>D</math> is a diagonal matrix whose diagonal values are in general complex.
The left and right singular vectors in the singular value decomposition of a normal matrix <math>A = U D V^*</math> differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.
Special casesEdit
Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues being unit modulus, real, and imaginary, respectively. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian, as their eigenvalues can be any complex number, in general. For example, <math display="block">A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}</math> is neither unitary, Hermitian, nor skew-Hermitian, because its eigenvalues are <math>2, (1\pm i\sqrt{3})/2</math>; yet it is normal because <math display="block">AA^* = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} = A^*A.</math>
ConsequencesEdit
Template:Math theorem Template:Math proof
The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies:
The diagonal entries of Template:Math are the eigenvalues of Template:Mvar, and the columns of Template:Mvar are the eigenvectors of Template:Mvar. The matching eigenvalues in Template:Math 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 Template:Math. Phrased differently: a matrix is normal if and only if its eigenspaces span Template:Math and are pairwise orthogonal with respect to the standard inner product of Template:Math.
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:
In general, the sum or product of two normal matrices need not be normal. However, the following holds:
In this special case, the columns of Template:Math are eigenvectors of both Template:Mvar and Template:Mvar and form an orthonormal basis in Template:Math. 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 definitionsEdit
It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let Template:Mvar be a Template:Math complex matrix. Then the following are equivalent:
- Template:Mvar is normal.
- Template:Mvar is diagonalizable by a unitary matrix.
- There exists a set of eigenvectors of Template:Mvar which forms an orthonormal basis for Template:Math.
- <math>\left\| A \mathbf{x} \right\| = \left\| A^* \mathbf{x} \right\|</math> for every Template:Math.
- The Frobenius norm of Template:Mvar can be computed by the eigenvalues of Template:Mvar: <math display="inline"> \operatorname{tr} \left(A^* A\right) = \sum_j \left| \lambda_j \right|^2 </math>.
- The Hermitian part Template:Math and skew-Hermitian part Template:Math of Template:Mvar commute.
- Template:Math is a polynomial (of degree Template:Math) in Template:Mvar.<ref group="lower-alpha">Proof: When <math>A</math> is normal, use Lagrange's interpolation formula to construct a polynomial <math>P</math> such that <math>\overline{\lambda_j} = P(\lambda_j)</math>, where <math>\lambda_j</math> are the eigenvalues of <math>A</math>.</ref>
- Template:Math for some unitary matrix Template:Mvar.<ref>Template:Harvp</ref>
- Template:Mvar and Template:Mvar commute, where we have the polar decomposition Template:Math with a unitary matrix Template:Mvar and some positive semidefinite matrix Template:Mvar.
- Template:Mvar commutes with some normal matrix Template:Mvar with distinctTemplate:Clarify eigenvalues.
- Template:Math for all Template:Math where Template:Mvar has singular values Template:Math and has eigenvalues that are indexed with ordering Template:Math.<ref>Template:Harvp</ref>
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.
Normal matrix analogyEdit
{{ safesubst:#invoke:Unsubst||date=__DATE__|$B= Template:Ambox }} It is occasionally useful (but sometimes misleading) to think of the relationships of special kinds of normal matrices as analogous to the relationships of the corresponding type of complex numbers of which their eigenvalues are composed. This is because any function (that can be expressed as a power series) of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition <math>VD V^*</math> is <math>VD^*V^*</math>, where <math>D</math> is the diagonal matrix of eigenvalues. Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on each corresponding pair of eigenvalues.
- The conjugate transpose is analogous to the complex conjugate.
- Unitary matrices are analogous to complex numbers on the unit circle.
- 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.
- Invertible matrices are analogous to non-zero complex numbers.
- The inverse of a matrix has each eigenvalue inverted.
- A uniform scaling matrix is analogous to a constant number.
- In particular, the zero matrix is analogous to 0, and
- the identity matrix is analogous to 1.
- An idempotent matrix is an orthogonal projection with each eigenvalue either 0 or 1.
- A normal involution has eigenvalues <math>\pm 1</math>.
As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mapping <math display="block">a + bi \mapsto \begin{bmatrix} a & b \\ -b & a \end{bmatrix} = a\, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + b\, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\,.</math> which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.
See alsoEdit
NotesEdit
<references group="lower-alpha" responsive="1"></references>
CitationsEdit
<references group="" responsive="1"></references>