Lesson 2.4 - Elementary Matrices
E_k ... E_2 E_1 A = I and ?
A = E_1^-1 E_2^-1 ... E_k^-1
According to the Definition of LU-Factorization, what is A equal to?
A = E_1^-1 E_2^-1 ... E_k^-1 U = LU
Equivalent Conditions: Condition 5
A can be written as the product of elementary matrices
What does it mean if A row-reduces to an upper triangular matrix U using only the row operation of adding a multiple of one row to another row below it?
A has an LU-Factorization
Equivalent Conditions: Condition 1
A is invertible
Equivalent Conditions: Condition 4
A is row-equivalent to Iₙ
Equivalent Conditions: Condition 3
Ax = O has only the trivial solution
Equivalent Conditions: Condition 2
Ax = b has a unique solution for every n x 1 column matrix b
The same elementary row operations used on a matrix A to reduce it to row-echelon form, when used on different elementary matrices, can be used in what expression?
B = E_3 E_2 E_1 A, or B = A reduced in row-echelon form
For elementary matrices & matrix multiplication, if the size of A is n x p, then what is the order of E?
E must have order n
According to the process of finding an LU-Factorization of a matrix, what does A = E_1^-1 E_2^-1 U give?
E_1^-1 E_2^-1 or L
How do you show that a matrix is the product of elementary matrices?
Find a sequence of elementary row ops that can be used to rewrite A in reduced row-echelon form From E_k ... E_2 E_1 A = I, solve for A to obtain A = E_1^-1 E_2^-1 ... E_k^-1
Once you have an LU-Factorization of matrix A, how can you solve the system of n linear equations in n variables Ax = b?
Forward Sub: write y = Ux & solve Ly = b for y Back Sub: solve Ux = y for x
In A = LU, what is L?
L is a lower triangular square matrix, which means all entries above the main diagonal are zero
According to the Definition of LU-Factorization, what is L equal to?
The product of the inverses of the elementary matrices used in the row reduction: L = E_1^-1 E_2^-1
According to the process of finding an LU-Factorization of a matrix, what does the row-reduced matrix of A give?
U
According to the Definition of LU-Factorization, what is U equal to?
U = E_k ... E_2 E_1 A
In A = LU, what is U?
U is an upper triangular square matrix, which means all entries below the diagonal are zero
Regarding elementary matrices, row multiplication of a matrix must be by what?
a nonzero constant
A Property of Invertible Matrices
a square matrix A is invertible if & only if it can be written as the product of elementary matrices (every invertible matrix can be written as the product of elementary matrices)
How do elementary matrices relate to invertible matrices?
all elementary matrices are invertible
The inverse of an elementary matrix is itself what?
an elementary matrix
Elementary Matrix
an n x n matrix is an elementary matrix when it can be obtained from the identity matrix Iₙ by a single elementary row operation
You can use elementary matrices & matrix multiplication to perform what on nonsquare matrices?
elementary row operations
When are Equivalent Conditions true?
if A is an n x n matrix
Elementary Matrices are Invertible
if E is an elementary matrix, then E^-1 exists & is an elementary matrix
Definition of LU-Factorization
if the n x n matrix A can be written as the product of a lower triangular matrix L & and upper triangular matrix U, then A = LU is an LU-factorization of A
Why is the identity matrix Iₙ elementary?
it can be obtained from itself by multiplying any of its rows by 1
If a matrix A is invertible, then how can it be written in relation to elementary matrices?
it can be written as the product of elementary matrices
What type of matrix is required for it to be elementary?
it must be square
Definition of Row Equivalence
let A & B be m x n matrices B is row-equivalent to A when there exists a finite number of elementary matrices E_1, E_2, ..., E_k s.t. B = E_k E_k-1 ... E_2 E_1 A
Representing Elementary Row Operations
let E be the elementary matrix obtained by performing an elementary row operation on I_m If that same elementary row operation is performed on an m x n matrix A, then the resulting matrix is the product EA
What is LU-Factorization used to solve?
linear systems Ax = b, in which the square matrix A is expressed as a product, A = LU
What does it mean if A can be written as the product of elementary matrices?
matrix A is invertible
What does it mean if U can be obtained from A using only the elementary row operation of adding a multiple of one row to another row below it?
matrix L is lower triangular (1's along the diagonal) the negative of each multiplier is in the same position as that of the corresponding zero in U below the main diagonal
A sequence of elementary matrices that reduces A to the identity also reduces what?
reduces the identity I to A^-1 Multiplying I = E_k ... E_2 E_1 A (on the right) by A^-1, A^-1 = E_k ... E_2 E_1 I E_k ... E_2 E_1 [ A I ] = [ I A^-1 ]
How do you find an LU-Factorization of a matrix?
row reduce the given matrix to upper triangular form while keeping track of the elementary matrices used for each row
According to the Definition of Row Equivalence, why is the order of multiplication important?
the elementary matrix immediately to the left of A corresponds to the row operation performed on the first
What is the inverse of an elementary matrix E?
the elementary matrix that converts E back to Iₙ
According to LU-Factorization and Ax = b, what is the column matrix x?
the solution of the original system since Ax = LUx = Ly = b
In terms of proofs, what does the phrase "if and only if" mean?
there are two parts
When is a matrix idempotent?
when A² = A
