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However, LU-factorization has the following advantages: Gaussian elimination and Gauss--Jordan elimination both use the augmented matrix [A|b], so b must be known. In contrast, LU-decomposition uses only matrix A, so once that factorization is complete, it can be applied to any vector b.
0:12 9:33 LU Decomposition - Shortcut Method - YouTube YouTube Start of suggested clip End of suggested clip And all elements above the main diagonal must be zero. For the upper triangular matrix all theMoreAnd all elements above the main diagonal must be zero. For the upper triangular matrix all the elements below the main diagonal must be zero. Notice for the upper triangular matrix the main diagonal.
This implies that for a square matrix: LUP always exists (We can use this to quickly figure out the determinant). If the matrix is invertible (the determinant is not 0), then a pure LU decomposition exists only if the leading principal minors are not 0.
1:04 8:25 LU Factorization Without Pivoting and Finding the Determinant - YouTube YouTube Start of suggested clip End of suggested clip So let's take a look the first step of Lu factorization involves row reducing so we want to make theMoreSo let's take a look the first step of Lu factorization involves row reducing so we want to make the bottom lower triangular left portion of the matrix zero all zeros.
Many complex matrix operations cannot be solved efficiently or with stability using the limited precision of computers. Matrix decompositions are methods that reduce a matrix into constituent parts that make it easier to calculate more complex matrix operations.
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0:26 8:22 Solve a System of Linear Equations Using LU Decomposition - YouTube YouTube Start of suggested clip End of suggested clip So if we substitute l u for a we would have the equation l times u times x equals b. Which we canMoreSo if we substitute l u for a we would have the equation l times u times x equals b. Which we can solve for x if we let u times x equal y. We would solve the equation l times y equals b for y.
0:57 10:38 LU Decomposition with Partial Pivoting | Lecture 26 - YouTube YouTube Start of suggested clip End of suggested clip We have to go down the first column of the matrix. And look for the largest element that occurs withMoreWe have to go down the first column of the matrix. And look for the largest element that occurs with the 6. Here. So we swap the first row in the second row.
LU decomposition is a better way to implement Gauss elimination, especially for repeated solving a number of equations with the same left-hand side. That is, for solving the equation Ax = b with different values of b for the same A.
LU decomposition is a better way to implement Gauss elimination, especially for repeated solving a number of equations with the same left-hand side. That is, for solving the equation Ax = b with different values of b for the same A.
0:26 8:22 Solve a System of Linear Equations Using LU Decomposition - YouTube YouTube Start of suggested clip End of suggested clip So if we substitute l u for a we would have the equation l times u times x equals b. Which we canMoreSo if we substitute l u for a we would have the equation l times u times x equals b. Which we can solve for x if we let u times x equal y. We would solve the equation l times y equals b for y.

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