Editing Normal matrix (section)

== Consequences ==
{{math theorem | name = Proposition | math_statement = A normal [[triangular matrix]] is [[diagonal matrix|diagonal]].}}
{{math proof | proof = Let {{mvar|A}} be any normal upper triangular matrix. Since
<math display="block">(A^* A)_{ii} = (A A^*)_{ii},</math>
using subscript notation, one can write the equivalent expression using instead the {{mvar|i}}th unit vector (<math>\hat \mathbf e_i</math>) to select the {{mvar|i}}th row and {{mvar|i}}th column:
<math display="block">\hat \mathbf e_i^\intercal \left(A^* A\right) \hat \mathbf e_i 
  = 
\hat \mathbf e_i^\intercal \left(A A^*\right) \hat \mathbf e_i.</math>
The expression
<math display="block">\left( A \hat \mathbf e_i \right)^* \left( A \hat \mathbf e_i\right) = \left( A^* \hat \mathbf e_i \right)^* \left( A^* \hat \mathbf e_i\right)</math>
is equivalent, and so is
<math display="block">\left \|A \hat \mathbf e_i \right\|^2 
  = 
\left \|A^* \hat \mathbf e_i \right \|^2,</math>
which shows that the {{mvar|i}}th row must have the same norm as the {{mvar|i}}th column.{{pb}}
Consider {{math|1=''i'' = 1}}. The first entry of row&nbsp;1 and column&nbsp;1 are the same, and the rest of column&nbsp;1 is zero (because of triangularity). This implies the first row must be zero for entries&nbsp;2 through {{mvar|n}}. Continuing this argument for row–column pairs&nbsp;2 through {{mvar|n}} shows {{mvar|A}} is diagonal. [[Q.E.D.]]}}

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

{{math theorem | name = Proposition | math_statement = A matrix {{mvar|A}} is normal if and only if there exist a [[diagonal matrix]] {{math|Λ}} and a [[unitary matrix]] {{mvar|U}}  such that {{math|1=''A'' = ''U''Λ''U''<sup>*</sup>}}.}}

The diagonal entries of {{math|Λ}} are the [[eigenvalue]]s of {{mvar|A}}, and the columns of {{mvar|U}} are the [[eigenvector]]s of {{mvar|A}}. The matching eigenvalues in {{math|Λ}} come in the same order as the eigenvectors are ordered as columns of {{mvar|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 {{math|'''C'''<sup>''n''</sup>}}. Phrased differently: a matrix is normal if and only if its [[eigenspace]]s span {{math|'''C'''<sup>''n''</sup>}} and are pairwise [[orthogonal]] with respect to the standard inner product of {{math|'''C'''<sup>''n''</sup>}}.

The spectral theorem for normal matrices is a special case of the more general [[Schur decomposition]] which holds for all square matrices. Let {{mvar|A}} be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, {{mvar|B}}. If {{mvar|A}} is normal, so is {{mvar|B}}. But then {{mvar|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: 

{{math theorem | name = Proposition | math_statement = A normal matrix is unitary if and only if all of its eigenvalues (its spectrum) lie on the unit circle of the complex plane.}}

{{math theorem | name = Proposition | math_statement = A normal matrix is [[self-adjoint]] if and only if its spectrum is contained in [[real number|<math>\R</math>]]. In other words: A normal matrix is [[Hermitian matrix|Hermitian]] if and only if all its eigenvalues are [[real number|real]].}}

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

{{math theorem | name = Proposition | math_statement = If {{mvar|A}} and {{mvar|B}} are normal with {{math|1=''AB'' = ''BA''}}, then both {{math|''AB''}} and {{math|''A'' + ''B''}} are also normal. Furthermore there exists a unitary matrix {{mvar|U}} such that {{math|''UAU''<sup>*</sup>}} and {{math|''UBU''<sup>*</sup>}} are diagonal matrices. In other words {{mvar|A}} and {{mvar|B}} are [[simultaneously diagonalizable]].}}

In this special case, the columns of {{math|''U''<sup>*</sup>}} are eigenvectors of both {{mvar|A}} and {{mvar|B}} and form an orthonormal basis in {{math|'''C'''<sup>''n''</sup>}}. 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.