Example:
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.
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
and for matrix multiplication,
The diagonal matrix diag(a_{1},...,a_{n}) is invertible if and only if the entries a_{1},...,a_{n} are all nonzero. In this case, we have
In particular, the diagonal matrices form a subring of the ring of all nbyn matrices.
Multiplying the matrix A from the left with diag(a_{1},...,a_{n}) amounts to multiplying the ith 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 ith column of A by a_{i} for all i.
Eigenvectors, eigenvalues, determinant
The eigenvalues of diag(a_{1},...,a_{n}) are a_{1},...,a_{n}. The unit vectors e_{1},...,e_{n} form a basis of eigenvectors. The determinant of diag(a_{1},...,a_{n}) is the product a_{1}...a_{n}.
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 nbyn 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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