Triangular Matrix - Description

Description

A matrix of the form

L= \begin{bmatrix} l_{1,1} & & & & 0 \\ l_{2,1} & l_{2,2} & & & \\ l_{3,1} & l_{3,2} & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ l_{n,1} & l_{n,2} & \ldots & l_{n,n-1} & l_{n,n} \end{bmatrix}

is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form

U = \begin{bmatrix} u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\ & u_{2,2} & u_{2,3} & \ldots & u_{2,n} \\ & & \ddots & \ddots & \vdots \\ & & & \ddots & u_{n-1,n}\\ 0 & & & & u_{n,n} \end{bmatrix}

is called an upper triangular matrix or right triangular matrix. The variable L (standing for lower or left) is commonly used to represent a lower triangular matrix, while the variable U (standing for upper) or R (standing for right) is commonly used for upper triangular matrix. A matrix that is both upper and lower triangular is diagonal.

Matrices that are similar to triangular matrices are called triangularisable.

The standard operations on triangular matrices preserve the triangular shape:

  • The sum of two upper triangular matrices is upper triangular.
  • The product of two upper triangular matrices is upper triangular.
  • The inverse of an invertible upper triangular matrix is upper triangular.
  • The product of an upper triangular matrix by a constant is an upper triangular matrix.

Together these facts mean that the upper triangular matrices form a Lie subalgebra of the Lie algebra of square matrices for any given size. The Lie algebra of all upper triangular matrices is often referred to as the Borel subalgebra, denoted . The analogous results hold for lower triangular matrices, so they also form a Lie subalgebra. However, note that the product of a lower triangular with an upper triangular matrix is not necessarily triangular.

Read more about this topic:  Triangular Matrix

Famous quotes containing the word description:

    “Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
    —Willard Van Orman Quine (b. 1908)

    “An intentional object is given by a word or a phrase which gives a description under which.”
    —Gertrude Elizabeth Margaret Anscombe (b. 1919)

    “To give an accurate description of what has never occurred is not merely the proper occupation of the historian, but the inalienable privilege of any man of parts and culture.”
    —Oscar Wilde (1854–1900)