Home > Diagonal matrix

n columns and n rows is diagonal if:

For example, the following matrix is diagonal:

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 I_n is 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.

1 Matrix operations

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

diag(a₁,...,a_n) + diag(b₁,...,b_n) = diag(a₁+b₁,...,a_n+b_n)

and for matrix multiplication,

diag(a₁,...,a_n) · diag(b₁,...,b_n) = diag(a₁b₁,...,a_nb_n).

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

diag(a₁,...,a_n)^-1 = diag(a₁^-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(a₁,...,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(a₁,...,a_n) amounts to multiplying the i-th column of A by a_i for all i.

2 Eigenvectors, eigenvalues, determinant

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

3 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 are diagonalizable matrixIn linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. if there exists an invertible matrix P such that P -1''AP is a diagonal matrix. If V is a finite- dimensional vector space, then a linear map T : V &ra.

Over the field of realIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may or complexThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher 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).