Lecture 2 Matrix Operations - Stanford University

Lecture 2

Matrix Operations

? transpose, sum & difference, scalar multiplication

? matrix multiplication, matrix-vector product

? matrix inverse

2�C1

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

2�C2

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

2�C3

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

2�C4

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

2�C5

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