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What is the inverse of a triangular matrix?

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I. INTRODUCTION TRIANGULAR matrix inversion (TMI) is a basic kernel in large and intensive scientific applications. Given its cubic complexity, several works addressed the design of efficient parallel algorithms for solving this problem. Apart the standard TMI algorithm consisting in solving n linear triangular systems of size n, n–1,…,1, a recursive algorithm, of same complexity, has been proposed by Heller in 1973 -. Our objective here is the design of a fast sequential algorithms based on Heller’s algorithm. The remainder of the paper is organized as follows. In section 2, we present the divide and conquer paradigm, then we detail a theoretical study on diverse sequential versions of Heller’s algorithm in section 3. Finally, we present in section 4 an experimental study.

Is the lower triangular matrix always invertible?

A triangular matrix is invertible if its diagonal entries are non-zero. Proposition A triangular matrix (upper or lower) is invertible if and only if all the entries on its main diagonal are non-zero.

Why is inverse of upper triangular matrix?

Inverse of Upper Triangular Matrix

To be invertible a square matrix must has determinant not equal to 0. Since, determinant of a upper triangular matrix is product of diagonals if it is nonzero, then the matrix is invertible.

What is lower triangular matrix with example?

In other words, a square matrix is lower triangular if all its entries above the main diagonal are zero. Example of a 3 × 3 lower triangular matrix: Diagonal matrices are both upper and lower triangular since they have zeroes above and below the main diagonal.

Is the inverse of a lower triangular matrix also lower triangular?

A triangular matrix (upper or lower) is invertible if and only if no element on its principal diagonal is 0. ... If the inverse L−1 of an lower triangular matrix L exists, then it is lower triangular.

image-What is the inverse of a triangular matrix?
image-What is the inverse of a triangular matrix?
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What is back substitution?

Mathwords: Back-Substitution. The process of solving a linear system of equations that has been transformed into row-echelon form or reduced row-echelon form.

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What do you think about multiplying a lower triangular matrix by a lower triangular matrix will the result be a lower triangular matrix?

The product of two lower triangular matrices is a lower triangular matrix. As a consequence, the product of any number of lower triangular matrices is a lower triangular matrix. If all the factor matrices are unit diagonal, then the resulting matrix is also unit diagonal.

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Which method we find lower triangular and upper triangular matrix?

A square matrix A can be decomposed into two square matrices L and U such that A = L U where U is an upper triangular matrix formed as a result of applying the Gauss Elimination Method on A, and L is a lower triangular matrix with diagonal elements being equal to 1. ; such that A = L U.Jul 20, 2021

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What is meant by Idempotent Matrix?

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily be a square matrix.

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Which condition is true for matrix A as lower triangular matrix?

A=[aij]n×n is lower triangular iff all entries above the diagonal vanish, i.e., if aij=0 for i<j.

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What is forward and backward substitution?

Forward substitution is the process of solving a system of linear algebraic equations (SLAE) Lx = y with a lower triangular coefficient matrix L. ... In, the process of solving a SLAE with a lower triangular coefficient matrix was named the back substitution.Feb 10, 2016

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When is the inverse of a lower triangular matrix invertible?

  • Proposition If a lower (upper) triangular matrix is invertible, then its inverse is lower (upper) triangular. Furthermore, each entry on the main diagonal of is equal to the reciprocal of the corresponding entry on the main diagonal of , that is, for . Let be a lower triangular matrix.

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What is an upper triangular matrix?

  • Definition A matrix is upper triangular if and only if whenever . Thus, in an upper triangular matrix all the elements below the main diagonal (i.e., those whose column index is less than the row index ) are zero.

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How do you know if a matrix is lower triangular?

  • A square matrix is said to be: lower triangular if all the elements above its main diagonal are zero; upper triangular if all the elements below its main diagonal are zero.

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What is the transpose of a triangular matrix?

  • The transpose of a triangular matrix is triangular The product of two triangular matrices is triangular A triangular matrix is invertible if its diagonal entries are non-zero The inverse of a triangular matrix is triangular Triangular matrices and echelon form Definition Formal definitions follow.

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When is the inverse of a lower triangular matrix invertible?When is the inverse of a lower triangular matrix invertible?

Proposition If a lower (upper) triangular matrix is invertible, then its inverse is lower (upper) triangular. Furthermore, each entry on the main diagonal of is equal to the reciprocal of the corresponding entry on the main diagonal of , that is, for . Let be a lower triangular matrix.

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What is a unit triangular matrix?What is a unit triangular matrix?

A unit triangular matrix is triangular matrix with 1s on the main diagonal. There are a few useful properties about products, inverses and determinants of triangular matrices [5]: The inverse of upper (lower) triangular matrix is upper (lower) triangular.

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How do you preserve upper triangularity of a matrix?How do you preserve upper triangularity of a matrix?

Upper triangularity is preserved by many operations: The sum of two upper triangular matrices is upper triangular. The product of two upper triangular matrices is upper triangular. The inverse of an upper triangular matrix, where extant, is upper triangular. The product of an upper triangular matrix and a scalar is upper triangular.

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What is the group of invertible triangular matrices?What is the group of invertible triangular matrices?

The set of invertible triangular matrices of a given kind (upper or lower) forms a group, indeed a Lie group, which is a subgroup of the general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).

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