Question

Factorizing a matrix A into a product of two or more matrices is a way to decompose a matrix into parts that are somehow more useful than the original matrix. For example, perhaps using the decomposed matrix can simplify more complex computations or can eliminate some redundant information. We will see some other factorizations later in the course. This exercise will walk you through LU factorization (or decomposition). That is, A will be decomposed into a product of a Lower triangular matrix times an Upper triangular matrix. - The matrix U is simply an echelon form of A produced by only using the row replacement operations E_ij(d). So, for some sequence of row replacement operations, U = E_kE_{k-1}...E_2E_1A or, equivalently, (E_kE_{k-1}...E_2E_1)^{-1}U = A. (Note that since only row replacement operations are used, entries along the main diagonal of U need not equal to 1.) - The choice for L is then clear. In particular, L = (E_kE_{k-1}...E_2E_1)^{-1} or, equivalently, L = E_1^{-1}E_2^{-1}...E_{k-1}^{-1}E_k^{-1}. - Note that (E_{ij}(d))^{-1} = E_{ij}(-d). (a) Using only a sequence of row replacement operation E_{ij}(d), find U which is the row echelon form of A = [[4, 3, -5], [-4, -5, 7], [8, 10, -8]] Keep track of each step and write the sequence of E_{ij}(d) row operations used (in order). (b) Write the sequence of inverse elementary matrices in the reverse order. (c) Find the product of these matrices to find L. (d) Verify that the product LU does equal the original matrix A.

          Factorizing a matrix A into a product of two or more matrices is a way to decompose a matrix into parts that are somehow more useful than the original matrix. For example, perhaps using the decomposed matrix can simplify more complex computations or can eliminate some redundant information. We will see some other factorizations later in the course. This exercise will walk you through LU factorization (or decomposition). That is, A will be decomposed into a product of a Lower triangular matrix times an Upper triangular matrix.
- The matrix U is simply an echelon form of A produced by only using the row replacement operations E_ij(d). So, for some sequence of row replacement operations,
U = E_kE_{k-1}...E_2E_1A or, equivalently, (E_kE_{k-1}...E_2E_1)^{-1}U = A.
(Note that since only row replacement operations are used, entries along the main diagonal of U need not equal to 1.)
- The choice for L is then clear. In particular,
L = (E_kE_{k-1}...E_2E_1)^{-1} or, equivalently, L = E_1^{-1}E_2^{-1}...E_{k-1}^{-1}E_k^{-1}.
- Note that
(E_{ij}(d))^{-1} = E_{ij}(-d).
(a) Using only a sequence of row replacement operation E_{ij}(d), find U which is the row echelon form of
A = [[4, 3, -5], [-4, -5, 7], [8, 10, -8]]
Keep track of each step and write the sequence of E_{ij}(d) row operations used (in order).
(b) Write the sequence of inverse elementary matrices in the reverse order.
(c) Find the product of these matrices to find L.
(d) Verify that the product LU does equal the original matrix A.

Factorizing a matrix A into a product of two or more matrices is a way to decompose a matrix into parts that are somehow more useful than the original matrix. For example, perhaps using the decomposed matrix can simplify more complex computations or can eliminate some redundant information. We will see some other factorizations later in the course. This exercise will walk you through LU factorization (or decomposition). That is, A will be decomposed into a product of a Lower triangular matrix times an Upper triangular matrix.
- The matrix U is simply an echelon form of A produced by only using the row replacement operations Eij(d). So, for some sequence of row replacement operations,
U = EkEk-1...E2E1A or, equivalently, (EkEk-1...E2E1)^-1U = A.
(Note that since only row replacement operations are used, entries along the main diagonal of U need not equal to 1.)
- The choice for L is then clear. In particular,
L = (EkEk-1...E2E1)^-1 or, equivalently, L = E1^-1E2^-1...Ek-1^-1Ek^-1.
- Note that
(Eij(d))^-1 = Eij(-d).
(a) Using only a sequence of row replacement operation Eij(d), find U which is the row echelon form of
A = [[4, 3, -5], [-4, -5, 7], [8, 10, -8]]
Keep track of each step and write the sequence of Eij(d) row operations used (in order).
(b) Write the sequence of inverse elementary matrices in the reverse order.
(c) Find the product of these matrices to find L.
(d) Verify that the product LU does equal the original matrix A.

Added by Sandra R.

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