Diagonal matrix

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In linear algebra, a diagonal matrix is a matrix (usually a square matrix) in which the entries outside the main diagonal (↘) are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (di,j) with n columns and n rows is diagonal if:

d_{i,j} = 0 \text{ if } i \ne j\ \forall i,j \in \{1, 2, \ldots, n\}

For example, the following matrix is diagonal:

1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -2\end{bmatrix}


1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -3\\
0 & 0 & 0\\
\end{bmatrix} or \begin{bmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 4 & 0& 0 & 0\\
0 & 0 & -3& 0 & 0\end{bmatrix}

However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well. Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular. The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.

Scalar matrix[edit]


    \lambda &       0 & 0       \\
          0 & \lambda & 0       \\
          0 &       0 & \lambda
  \end{bmatrix} \equiv \lambda \boldsymbol{I}_3


For an abstract vector space V (rather than the concrete vector space K^n), or more generally a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices, the analog of scalar matrices are scalar transformations. Formally, scalar multiplication is a linear map, inducing a map R \to \operatorname{End}(M), (send a scalar λ to the corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R-algebra. For vector spaces, or more generally free modules M \cong R^n, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL(V), where they are denoted by Z(V), follow the usual notation for the center.

Matrix operations[edit]

The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a1, ..., an) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1, ..., an. Then, for addition, we have


and for matrix multiplication,






Operator matrix in eigenbasis[edit]

As explained in determining coefficients of operator matrix, there is a special basis, e1, ..., en, for which the matrix takes the diagonal form. Being diagonal means that all coefficients a_{i,j} but a_{i,i} are zeros in the defining equation A \vec e_j = \sum a_{i,j} \vec e_i, leaving only one term per sum. The surviving diagonal elements, a_{i,i}, are known as eigenvalues and designated with \lambda_i in the equation, which reduces to A \vec e_i = \lambda_i \vec e_i. The resulting equation is known as eigenvalue equation[1] and used to derive the characteristic polynomial and, further, eigenvalues and eigenvectors.


Other properties[edit]


The adjugate of a diagonal matrix is again diagonal.



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 map by a diagonal matrix.



Operator theory[edit]



See also[edit]

