Matrix Primer - Stanford University
Lecture 2 Matrix Operations
? transpose, sum & difference, scalar multiplication ? matrix multiplication, matrix-vector product ? matrix inverse
2?1
Matrix transpose
transpose of m ? n matrix A, denoted AT or A, is n ? m matrix with
AT ij = Aji
rows and columns of A are transposed in AT
0 4 T example: 7 0 =
31
073 401
.
? transpose converts row vectors to column vectors, vice versa ? AT T = A
Matrix Operations
2?2
Matrix addition & subtraction
if A and B are both m ? n, we form A + B by adding corresponding entries
0 4 1 2 1 6
example: 7 0 + 2 3 = 9 3
31
04
35
can add row or column vectors same way (but never to each other!)
matrix subtraction is similar:
16 93
-I =
06 92
(here we had to figure out that I must be 2 ? 2)
Matrix Operations
2?3
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 + BT
Matrix Operations
2?4
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:
1 6 -2 -12
(-2) 9 3 = -18 -6
60
-12 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?5
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