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Diagonal matrix

In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. The diagonal entires may or may not be zero. Thus, the matrix D = (di,j) is diagonal if:

<math>d_{i,j} = 0 \mbox{ if } i \ne j </math>

Example:

<math>\begin{bmatrix}
1 & 0 \\ 0 & 4 \end{bmatrix}</math>

Any diagonal matrix is also a symmetric matrix, a triangular matrix[?], and (if the entries come from the field R or C) also a normal matrix. The identity matrix In is diagonal.

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.

Eigenvectors, eigenvalues, determinant

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.

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 is similar to a diagonal matrix if and only if it has n linearly independent eigenvectors. These matrices are called diagonalizable.

Over the field of real or complex numbers, more is true: every normal matrix is unitarily similar to a diagonal matrix (the spectral theorem), and every matrix is unitarily equivalent[?] to a diagonal matrix with nonnegative entries (the singular value decomposition).



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