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In linear algebra, a diagonal matrix is 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 = (d_{i,j}) with
n columns and n rows is diagonal if:## Matrix operations

## Other properties

## Uses

## See also

## External links

## References

- $d\_\{i,j\}\; =\; 0\; mbox\{\; if\; \}\; i\; ne\; j\; qquad\; forall\; i,j\; in$

For example, the following matrix is diagonal:

- $begin\{bmatrix\}$

The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form d_{i,i} possibly non-zero; for example,

- $begin\{bmatrix\}$

However, in the remainder of this article we will consider only square matrices.

Any 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 I_{n} and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.

A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Its effect on a vector is scalar multiplication by λ. The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.

The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a_{1},...,a_{n}) for a diagonal matrix whose diagonal entries starting in the upper left corner are a_{1},...,a_{n}. Then, for addition, we have

- diag(a
_{1},...,a_{n}) + diag(b_{1},...,b_{n}) = diag(a_{1}+b_{1},...,a_{n}+b_{n})

and for matrix multiplication,

- diag(a
_{1},...,a_{n}) · diag(b_{1},...,b_{n}) = diag(a_{1}b_{1},...,a_{n}b_{n}).

The diagonal matrix diag(a_{1},...,a_{n}) is invertible if and only if the entries a_{1},...,a_{n} are all non-zero. In this case, we have

- diag(a
_{1},...,a_{n})^{-1}= diag(a_{1}^{-1},...,a_{n}^{-1}).

In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices.

Multiplying the matrix A from the left with diag(a_{1},...,a_{n}) amounts to multiplying the i-th row of A by a_{i} for all i; multiplying the matrix A from the right with diag(a_{1},...,a_{n}) amounts to multiplying the i-th column of A by a_{i} for all i.

The eigenvalues of diag(a_{1}, ..., a_{n}) are a_{1}, ..., a_{n} with associated eigenvectors of e_{1}, ..., e_{n}, where the vector e_{i} is all zeros except a one in the ith row. The determinant of diag(a_{1}, ..., a_{n}) is the product a_{1}...a_{n}.

The adjugate of a diagonal matrix is again diagonal.

A square matrix is diagonal if and only if it is triangular and normal.

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.

In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that XAX^{-1} is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.

Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA^{*} = A^{*}A then there exists a unitary matrix U such that UAU^{*} is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV^{*} is diagonal with positive entries.

- Anti-diagonal matrix
- Bidiagonal matrix
- Tridiagonal matrix
- Banded matrix
- Diagonally dominant matrix
- Toeplitz matrix
- Circulant matrix

- Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).

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Last updated on Saturday July 05, 2008 at 02:39:39 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday July 05, 2008 at 02:39:39 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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