diagonal matrix

Definition Let A be a square matrixMathworldPlanetmath (with entries in any field). If all off-diagonal entries of A are zero, then A is a diagonal matrixMathworldPlanetmath.

From the definition, we see that an n×n diagonal matrix is completely determined by the n entries on the diagonal; all other entries are zero. If the diagonal entries are a1,a2,,an, then we denote the corresponding diagonal matrix by



  1. 1.

    The identity matrixMathworldPlanetmath and zero matrixMathworldPlanetmath are diagonal matrices. Also, any 1×1 matrix is a diagonal matrix.

  2. 2.

    A matrix A is a diagonal matrix if and only if A is both an upper and lower triangular matrixMathworldPlanetmath.


  1. 1.

    If A and B are diagonal matrices of same order, then A+B and AB are again a diagonal matrix. Further, diagonal matrices commute, i.e., AB=BA. It follows that real (and complex) diagonal matrices are normal matricesMathworldPlanetmath.

  2. 2.

    A square matrix is diagonal if and only if it is triangular and normal (see this page (http://planetmath.org/TheoremForNormalTriangularMatrices)).

  3. 3.

    The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a diagonal matrix A=diag(a1,,an) are a1,,an. Corresponding eigenvectorsMathworldPlanetmathPlanetmathPlanetmath are the standard unit vectors in n. For the determinantMathworldPlanetmath, we have detA=a1a2an, so A is invertiblePlanetmathPlanetmath if and only if all ai are non-zero. Then the inverse is given by

  4. 4.

    If A is a diagonal matrix, then the adjugate of A is also a diagonal matrix.

  5. 5.

    The matrix exponentialMathworldPlanetmath of a diagonal matrix is


More generally, every analytic functionMathworldPlanetmath of a diagonal matrix can be computed entrywise, i.e.:



Diagonal matrices are also sometimes called quasi-scalar matrices [1].


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath Theory, Dover publications, 1980.
  • 2 Wikipedia, http://www.wikipedia.org/wiki/Diagonal_matrixdiagonal matrix.
Title diagonal matrix
Canonical name DiagonalMatrix
Date of creation 2013-03-22 13:43:32
Last modified on 2013-03-22 13:43:32
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 15-00
Classification msc 15A57
Synonym quasi-scalar matrix
Synonym quasi-scalar matrices
Synonym diagonal matrices
Related topic DiagonalizationLinearAlgebra