First row becomes first column.
First column becomes first row.
Etc.
1 4 transposed = 1 6 3
6 1 4 1 5
3 5
Add up all the stuff in the same row/column.
1 2 + 5 6 = 6 8
3 4 7 8 10 12
Similar to Addition.
7 2 - 5 6 = 2 -4
3 4 7 8 -4 12
Similar to Addition.
1 * 4 = 4
6 1 6
3 5 15
1 2 * 5 6 = 5 12
3 4 7 8 21 32
Multiply the scalar by each value.
3 * 3 5 4 = 9 15 12
6 1 2 18 3 6
Multiplies two vectors into one number.
Multiply corresponding row/columns. Then sum those.
1 * 6 = 35
7 2
3 5
1*6 + 7*2 + 3*5 = 6 + 14 + 15
= 20 + 15
= 35
To multiply A * B = C:
-
Treat each row in A as a vector.
-
Multiply that row vector by B.
-
Place the result in same row in C.
3 5 4 6 * 2 = 78
8 3 6 1 6 57
3
5
3 5 4 6 * 2 = 3 * 2 = 3*2 + 5*6 + 4*3 + 6*5
6 5 6
3 4 3
5 6 5
= 6 + 30 + 12 + 30
= 60 + 18
= 78
8 3 6 1 * 2 = 8 * 2 = 8*2 + 3*6 + 6*3 + 1*5
6 3 6
3 6 3
5 1 5
= 16 + 18 + 18 + 5
= 17 + 20 + 20
= 17 + 40
= 57
To multiply A * B = C.
-
Do matrix-vector multiplication on A.row_1 * B.column_1.
-
Place result in C.row_1,column_1.
-
Do matrix-vector multiplication on A.row_1 * B.column_2.
-
Place result in C.row_1,column_2.
-
Etc.
3 5 4 6 * 2 6 = 69 60
8 3 6 1 7 2 47 86
1 5
4 2
3 5 4 6 * 2 = 3 * 2 = 3*2 + 5*7 + 4*1 + 6*4
7 5 7
1 4 1
4 6 4
= 6 + 35 + 4 + 24
= 10 + 35 + 24
= 40 + 29
= 69
3 5 4 6 * 6 = 3 * 6 = 3*6 + 5*2 + 4*5 + 6*2
2 5 2
5 4 5
2 6 2
= 18 + 10 + 20 + 12
= 30 + 10 + 20
= 60
8 3 6 1 * 2 = 8*2 + 3*7 + 6*1 + 1*4
7
1
4
= 16 + 21 + 6 + 4
= 20 + 27
= 47
8 3 6 1 * 6 = 8*6 + 3*2 + 6*5 + 1*2
2
5
2
= 48 + 6 + 30 + 2
= 50 + 36
= 86
A(n by m) * B(m by p) = C(n by p)
Associative.
- (AB)C = A(BC)
Not Commutative.
- AB != BA
Multiplies two Vectors, A * B = C.
Result is a Matrix.
Steps:
-
Treat A and B as Matrices.
-
Perform A * B.transpose.
- See Matrix-Matrix Multiplication.
3 outer_product 5 = 3 * 5 6 3 8 = 15
2 6 2 12
6 3 6 18
2 8 2 16
3*5 = 15
2*6 = 12
6*3 = 18
2*8 = 16
It's like absolute value for vectors.
It's the square root of the sum of the squares of each element.
4 euclidean_norm = sqrt(4^2 + 3^2 + (-5)^2 + 1^2 + (-2)^2)
3
-5
1
-2
= sqrt(16 + 9 + 25 + 1 + 4)
= sqrt(20 + 25 + 10)
= sqrt(55)
= 7.4162
Think of this in scalar terms: c * 1 = c.
1 is the scalar multiplication identity because multiplying it by c doesn't change c's value.
The identity matrix is the same thing for matrices.
The identity matrices always follow the same pattern as the one in these examples.
9 3 5 * 1 0 0 = 9 3 5
4 1 2 0 1 0 4 1 2
0 0 1
1 0 0 * 9 3 5 = 9 3 5
0 1 0 4 1 2 4 1 2
0 0 1
Think of this in scalar terms: c * 1/c = 1.
7 * 1/7 = 7.
1/c is the scalar inverse because multiplying it by c gives 1.
The inverse matrix a similar thing for matrices.
Multiplying a matrix A by its inverse gives the identity matrix.
A(A.inverse) = (A.inverse)A = Identity
The inverse matrix does not exist for every matrix.
The matrix A must be a square matrix.