Lecture 2 Matrix Operations - Stanford University
Lecture 2
Matrix Operations
? transpose, sum & difference, scalar multiplication
? matrix multiplication, matrix-vector product
? matrix inverse
2C1
Matrix transpose
transpose of m n matrix A, denoted AT or A, is n m matrix with
A
T
ij
= Aji
rows and columns of A are transposed in AT
?T
?
0 4
0 7 3
.
example: ? 7 0 ? =
4 0 1
3 1
? transpose converts row vectors to column vectors, vice versa
? A
T T
=A
Matrix Operations
2C2
Matrix addition & subtraction
if A and B are both m n,
? ?
?
1
0 4
example: ? 7 0 ? + ? 2
0
3 1
we form A + B by adding corresponding entries
?
? ?
1 6
2
3 ?=? 9 3 ?
3 5
4
can add row or column vectors same way (but never to each other!)
1 6
0 6
matrix subtraction is similar:
?I =
9 3
9 2
(here we had to figure out that I must be 2 2)
Matrix Operations
2C3
Properties of matrix addition
? commutative: A + B = B + A
? associative: (A + B) + C = A + (B + C), so we can write as A + B + C
? A + 0 = 0 + A = A; A ? A = 0
? (A + B)T = AT + B T
Matrix Operations
2C4
Scalar multiplication
we can multiply a number (a.k.a. scalar ) by a matrix by multiplying every
entry of the matrix by the scalar
this is denoted by juxtaposition or , with the scalar on the left:
?
? ?
?
?2 ?12
1 6
(?2) ? 9 3 ? = ? ?18 ?6 ?
?12
0
6 0
(sometimes you see scalar multiplication with the scalar on the right)
? ( + )A = A + A; ()A = ()(A)
? (A + B) = A + B
? 0 A = 0; 1 A = A
Matrix Operations
2C5
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