# What is matrix triangulation?

## What is matrix triangulation?

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

## What is the best way to multiply a chain of matrices?

Take the sequence of matrices and separate it into two subsequences. Find the minimum cost of multiplying out each subsequence. Add these costs together, and add in the cost of multiplying the two result matrices.

**How do you solve matrix chain multiplication problems?**

For example, suppose A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × 60 matrix. Then, (AB)C = (10×30×5) + (10×5×60) = 1500 + 3000 = 4500 operations A(BC) = (30×5×60) + (10×30×60) = 9000 + 18000 = 27000 operations. Clearly the first parenthesization requires less number of operations.

**What is matrix chain multiplication example?**

Example of Matrix Chain Multiplication. Example: We are given the sequence {4, 10, 3, 12, 20, and 7}. The matrices have size 4 x 10, 10 x 3, 3 x 12, 12 x 20, 20 x 7. We need to compute M [i,j], 0 ≤ i, j≤ 5. We know M [i, i] = 0 for all i.

### Where is matrix chain multiplication used?

Matrix Chain Multiplication is one of the optimization problem which is widely used in graph algorithms, signal processing and network industry [1–4]. We can have several ways to multiply the given number of matrices because the matrix multiplication is associative.

### Why Parenthesization is important in matrix multiply?

Matrix Chain Multiplication Problem can be stated as “find the optimal parenthesization of a chain of matrices to be multiplied such that the number of scalar multiplication is minimized”. Number of ways for parenthesizing the matrices: There are very large numbers of ways of parenthesizing these matrices.

**What is time complexity of matrix chain multiplication?**

As before, if we have n matrices to multiply, it will take O(n) time to generate each of the O(n2) costs and entries in the best matrix for an overall complexity of O(n3) time at a cost of O(n2) space.

**What is the triangular formula?**

The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles, whether it is a scalene triangle, an isosceles triangle, or an equilateral triangle.

#### How do you find the triangulation of matrices?

Triangulation of matrices. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their union. If the union does not form a basis of , then we would add some extra elements from the canonical basis of to that union to form a basis of .

#### Is Matrix Triangulation Gaussian elimination?

Matrix triangulation (Bareiss method) It’s also Gaussian elimination as it’s a method of successive elimination of variables, when with the help of elementary transformations the equation systems is reduced to a row echelon (or triangular) form, in which all other variables are placed (starting from the last).

**How to find a triangular matrix similar to a triangularizable matrix?**

Suppose that A is some triangularizable matrix in M n ( R). The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their union.

**What are the applications of matrix multiplication?**

Although there are many applications of matrices, essentially, multiplication of matrices is an operation in linear algebra. The linear mapping which includes scalar addition and multiplication is represented by matrix multiplication. One can also find a wide range of algorithms on meshes.