Diagonal matrix

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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 = (di,j) with n columns and n rows is diagonal if:

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

For example, the following matrix is diagonal:

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

The term diagonal matrix may sometimes refer to a rectangular m-by-n matrix in case only the entries of the form ai,i are non-zero. However, in 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 In 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.

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[edit] Matrix operations

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

diag(a1,...,an) + diag(b1,...,bn) = diag(a1+b1,...,an+bn)

and for matrix multiplication,

diag(a1,...,an) · diag(b1,...,bn) = diag(a1b1,...,anbn).

The diagonal matrix diag(a1,...,an) is invertible if and only if the entries a1,...,an are all non-zero. In this case, we have

diag(a1,...,an)-1 = diag(a1-1,...,an-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(a1,...,an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1,...,an) amounts to multiplying the i-th column of A by ai for all i.

[edit] Other properties

The eigenvalues of diag(a1, ..., an) are a1, ..., an. The unit vectors e1, ..., en form a basis of eigenvectors. The determinant of diag(a1, ..., an) is the product a1...an.

The adjugate of a diagonal matrix is again diagonal.

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

[edit] Uses

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.

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