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[[ماتریس همسازه]] برای یک ماتریس قطری به صورت زیر است:
<math>A=\begin{bmatrix}
a & 0 & 0\\
== خواص دیگر ==
The [[eigenvalueویژه‌مقدار]]sهای ofماتریس diag(''a''<sub>1</sub>,قطری ...,که ''a''<sub>''n''</sub>)قطر areاصلی آن''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> withباشد associatedبرابر [[eigenvectors]]است ofبا ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>, where the vector ''e''<sub>''i''</sub> is all zeros except a one in the ''i''th row. The [[determinant]] of diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>) is the product ''a''<sub>1</sub>...''a''<sub>''n''</sub>.
The [[adjugate]] of a diagonal matrix is again diagonal.
A square matrix is diagonal if and only if it is triangular and [[Normal_matrix|normal]].
== کاربردها ==
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or [[linear operator|linear map]] by a diagonal matrix.
مزدوج هر ماتریس قطری نیز خود یک ماتریس قطری است.
In fact, a given ''n''-by-''n'' matrix ''A'' is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix ''X'' such that ''X<sup>-1</sup>AX'' is diagonal) if and only if it has ''n'' [[linearly independent]] eigenvectors. Such matrices are said to be [[diagonalizable matrix|diagonalizable]].
یک ماتریس مربعی قطری است اگر و فقط اگر [[ماتریس مثلثی|مثلثی]] و [[ماتریس نرمال|نرمال]] باشد.
Over the [[field (mathematics)|field]] of [[real number|real]] or [[complex number|complex]] numbers, more is true. The [[spectral theorem]] says that every [[normal matrix]] is [[similar matrix|unitarily similar]] to a diagonal matrix (if ''AA''<sup>*</sup> = ''A''<sup>*</sup>''A'' then there exists a [[unitary matrix]] ''U'' such that ''UAU''<sup>*</sup> is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix ''A'', there exist unitary matrices ''U'' and ''V'' such that ''UAV''<sup>*</sup> is diagonal with positive entries.
== جستارهای وابسته==
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