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

diag(a1,,an)=(a10000a20000a30000an).

Examples

  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.

Properties

  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

    (diag(a1,,an))-1=diag(1/a1,,1/an).
  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

    ediag(a1,,an)=diag(ea1,,ean).

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

f(diag(a11,a22,,ann))=diag(f(a11),f(a22),,f(ann))

Remarks

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

References

  • 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