lecture 2 - matrices Flashcards
what is a matrix?
table of n rows and m columns
A = a1,1 a1,2 … a1,n
a2,1 a2,2 .. a2,n
… … … …
am,1 am,2 .. am,n
How is a matrice denoted?
R ( real number) (m,n) with m being number of columns, and n being number of rows.
What is a square matrix?
If R(n,n)
As in if m = n ( no. of columns and rows is the same)
Otherwise is a rectangular matrix
What does a(i) and a^j mean?
a(i) = ith row vector
a^j = jth column vector
What is transpose?
denoted as A^T
elements in a matrix shifted by 90 degrees on square matrices
1 2
3 5
»>
1 3
2 5
What is symmetry?
if element a(i,j) = a(j,i)
- imagine a symmetrical line though the diagonal.
- follws that A^T = A
What is diagonal?
elements where row index = column index formatted into an ordered tuple.
e.g.
1 2 3
3 5 1
2 4 5
diagonal = (1,5,5)
What is a trace?
symbol as tr(A)
sum of diagonal elements
e.g.
1 2 3
3 5 1 = A
2 4 5
tr(A) = 1 + 5 + 5 = 11
What is the null matrix?
denoted O
- all elements in a matrix is 0
e.g.
0 0 0
0 0 0
0 0 0
What is the identity matrix?
Denoted I
square matrix whose diagonal elements are 1s, with rest 0
e.g.
1 0 0
0 1 0
0 0 1
what is sum of matrices?
for element c(i,j) = a(i,j) + b(i,j)
e.g.
2 0 3 0 5 0
4 8 + 4 9 = 8 17
what is the scalar multiplication to a matrice?
element c(i,j) = lambda * a(i,j)
e.g.
2 0 4 0
3 8 * 2 = 8 16
what is a matrix product?
product of matrix A and B where C = AB
Take the row vector of A and multiply by the column vector of B.
> matrices must have at least the same amount of column in B as rows in A
=
(a(1) * b(1) a(1)b(2))
(a(2)b(1) a(2)b(2))
example of matrix product?
A =
2 7 3 1
5 0 4 1
B =
1 2
2 5
8 0
2 2
For first element a(1)b(1) is:
(2 * 1) + (72) + (38) + (12) = 42
what is transpose product?
Rule that
(AB)^T =B^T * A^T
