Chapter 2 Flashcards
Zero matrix
a m x n matrix of zeros denoted 0
triangular matrix
n x n matrix where entries above or below the main diagonal are zero
diagonal matrix
n x n matrix where all entries off the main diagonal are zero
identity matrix
n x n diagonal matrix with 1’s along the main diagonal, denoted I_n
When are two matrices equal?
A = B when A and B have the same size and corresponding entries
How are two matrices added?
(A + B): add entry-wise (must have the same size)
How is a matrix subjected to scalar multiplication?
(kA) scale each entry by k
-A = (-1)A
A-B = A + (-B)
How is a matrix transposed?
A^t is the matrix whose columns are the rows of A
ij entry of A = ji entry of A^t
How is matrix multiplication performed?
If A is an m x n matrix and B is an n x p matrix with columns b1, … , bp, then
AB = A[b1 b2 … bp] = [Ab1 Ab2 … Abp]
(Across and Down rule:
[AB]ij = (row i of A)*(col j of B)
What are the properties of matrix addition?
Let A, B, and C be matrices with sizes such that the indicated operations are defined. The following hold:
i) A + B = B + A
ii) A + ( B + C ) = ( A + B ) + C
iii) A + 0 = A
What are the properties of scalar multiplication of matrices?
Let A, B, and C be matrices with sizes such that the indicated properties are defined. Let r, s be scalars. The following hold:
iv) r(A + B) = rA + rB
v) (r + s)A = rA + sA
vi) r(sA) = (rs)A
What are the properties of matrix multiplication?
Let A, B, and C be matrices with sizes such that the indicated operations are defined. Let r, s be scalars. The following hold:
vii) A(BC) = (AB)C
viii) A(B + C) = AB + AC
ix) (A + B)C = AC + BC
x) r(AB) = (rA)B = A(rB)
xi) I_m * A = A = A* I_m (where I_m represents the m x m identity matrix)
What are the properties of matrix transposition?
Let A, B, and C be matrices with sizes such that the indicated operations are defined. Let r, s be scalars. The following hold:
xii) (A^T)^T = A
xiii) (A + B)^T = A^T + B^T
xiv) (r*A)^T = r * A^T
xv) (AB)^T = (B^T)(A^T)
There are some possible outcomes in matrix arithmetic that don’t occur in usual scalar arithmetic. What are they?
The following are possible for matrices A, B, and C:
1: AB is not equal to BA
2: AB = AC but B is not equal to C
3: AB = 0 but A and B are not equal to zero
When is a matrix B the inverse of a matrix A?
If A is a square matrix, then B is called the inverse of A if AB = BA = I
Additionally, the following phrases are equivalent: has an inverse, invertible, nonsingular
If A has an inverse, then it is unique and is denoted A^-1
What are the properties of inverses? (Theorem 2.6)
If A and B are invertible n x n matrices, then:
1: (A^-1)^-1 = A
2: (AB)^-1 = (B^-1)(A^-1) (socks-shoes property)
3: (A^T)^-1 = (A^-1)^T
Theorem 2.5
If the n x n matrix A is invertible, then the system Ax = b has a unique solution for each b in R^n, namely x = (A^-1)b
How can the inverse of a 2 x 2 matrix be found? (Theorem 2.4)
Let A = [ a, b; c, d]. Then A is invertible if and only if ad-bc is not equal to 0. If A is invertible, then
A^-1 = (1/(ad-bc))*[d, -b; -c, a]
(the number ad-bc is called the determinant of A)
How can the inverse of a matrix be found in general?
An n x n matrix A is invertible if and only if A is row equivalent to I, and in this case, any sequence of elementary row operations that reduces A to I, also transforms I into A^-1
In other words, the theorem says:
A is invertible if and only if [ A | I ] is row equivalent to [ I | A^-1 ]
Elementary matrix
One obtained by performing a single elementary row operation on I
Facts about elementary matrices
1: if E is an elementary matrix, then EA is the matrix that results if the elementary row operation that produced E is performed on A. (We can perform row operations via matrix multiplication)
2: each elementary matrix is invertible, and E and E^-1 represent “inverse operations”
What is the endpoint of LU factorization?
Factoring the m x n matrix A into the matrix product LU where:
1: L is a lower triangular m x m matrix with 1’s on the main diagonal
2: U is an RE form of A
What is the algorithm for LU factorization?
1: Reduce A to RE form U using ONLY (lower) row replacement operations (no scaling or interchanging!) if possible.
2: Keep track of each replacement operation
3: Form L by applying the inverse operations, in reverse order, to the identity I
Why do we insist on only using certain row replacement operations in our row reduction?
All other row operations mess up L as a unit lower triangular matrix
Does every matrix A have an LU factorization?
Nah
Why is LU factorization useful?
If solving the system Ax = b for repeated values of b, the LU solution method involves fewer individual addition/multiplication computations than multiplication by A^-1 (if it exists) or repeated elementary row reduction.
