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In this article, let us explore the different types of triangular matrices including upper triangular matrix and lower triangular matrix, their definitions, and their properties. We will also solve some examples based on a triangular matrix for a better understanding of the concept. A triangular matrix is a special kind of square matrix in the set of matrices. There are two types of triangular matrices: lower triangular matrix and upper triangular matrix.
There are different types of triangular matrices that we study. Given below is a list of some special types of triangular matrices:. In the following sections, we will mainly explore two types of triangular matrices, namely an upper and lower triangular matrix. This implies that all elements below the main diagonal of a square matrix are zero in an upper triangular matrix.
An example of an upper triangular matrix is given below:. This implies that all elements above the main diagonal of a square matrix are zero in a lower triangular matrix. An example of a lower triangular matrix is given below:. Since we have understood the meaning of a triangular matrix, let us go through some of its important properties. Given below is a list of the properties of a triangular matrix:. Example 1: Identify if the given matrix is a triangular matrix.
Also, identify its type. Therefore, the given matrix is a lower triangular matrix as the element above the main diagonal is zero. Example 2: Find the values of 'a' and 'b' in the given matrix B such that B is a strictly upper triangular matrix. Assuming B is a strictly upper triangular matrix, the elements below the diagonal are zero and the elements of the main diagonal are zero.
The given matrix is a triangular matrix lower as its elements above the diagonal are all zeros. A triangular matrix is a special type of square matrix in linear algebra whose elements below and above the diagonal appear to be in the form of a triangle. If a matrix is both lower and upper triangular, then all its non-diagonal elements are equal to zeros.
In this case, it is called a diagonal matrix. Take a look at this post: math. Add a comment. Sorted by: Reset to default. Highest score default Date modified newest first Date created oldest first. Michael0x2a I'm trying to get the inverse of a triangular matrix, not a square matrix. How back substitution should help me in obtaining the inverses of the triangulars? By defintion, triangular matrices are square.
Moreover, only square matrices have inverses. Don't invert it if you can. It's one of the basic commandments of numerical linear algebra. Fanfan Fanfan 1, 5 5 silver badges 7 7 bronze badges. Greg Rogers Greg Rogers Guilherme Barros Guilherme Barros 11 1 1 bronze badge.
Charlie Martin Charlie Martin k 23 23 gold badges silver badges bronze badges. Inverting a triangular matrix is not half the contents of a course in numerical analysis. Inverting a triangular matrix is trivial, and the naive algorithm is stable. Gotta do row pivoting. Naive won't be stable. Pivoting via, say, Gaussian elimination puts a linear system into triangular form which is then solved with backsubstitution.
As for stability see, for example, the book Accuracy and Stability of Numerical Algorithms by Nicholas Higham, page of the second edition. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog.
Another method is as follows. An invertible upper triangular matrix has the form A=D(I+N) where D is diagonal (with the same diagonal entries as A) and N is. The upper triangular matrix has all the elements below the main diagonal as zero. Also, the matrix which has elements above the main diagonal as zero is. A matrix whose number of rows is equal to the number of columns can only be represented as triangular matrix. A matrix which are similar to triangular matrix as.