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A matrix is set of Numbers."
This Matrix has 1 row and 3 columns:
C = |
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The Dimension of the matrix is (1x3).
This matrix has 2 rows and 3 columns:
C = |
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The dimension of the matrix is (2x3).
A Square Matrix is a matrix with the same number of rows and columns.
An n-by-n matrix is known as a square matrix of order n.
A 2-by-2 matrix (Square matrix of order 2):
C = |
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A 4-by-4 matrix (Square matrix of order 4):
C = |
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A Diagonal Matrix has values on the diagonal entries, and zero on the rest:
C = |
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A Scalar Matrix has equal diagonal entries and zero on the rest:
C = |
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The Identity Matrix has 1 on the diagonal and 0 on the rest.
This is the matrix equivalent of 1. The symbol is I.
I = |
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If you multiply any matrix with the identity matrix, the result equals the original.
The Zero Matrix (Null Matrix) has only zeros.
C = |
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Matrices are Equal if each element correspond:
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The Negative of a matrix is easy to understand:
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In linear algebra, the most simple math object is the Scalar:
const scalar = 1;
Another simple math object is the Array:
const array = [ 1, 2, 3 ];
Matrices are 2-dimensional Arrays:
const matrix = [ [1,2],[3,4],[5,6] ];
Vectors can be written as Matrices with only one column:
const vector = [ [1],[2],[3] ];
Vectors can also be written as Arrays:
const vector = [ 1, 2, 3 ];
Programming matrix operations in JavaScript, can easily become a spaghetti of loops.
Using a JavaScript library will save you a lot of headache.
One of the most common libraries to use for matrix operations is called math.js.
It can be added to your web page with one line of code:
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjs/9.3.2/math.js"></script>
If two matrices have the same dimension, we can add them:
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const mA = math.matrix([[1, 2], [3, 4], [5, 6]]);
const mB = math.matrix([[1,-1], [2,-2], [3,-3]]);
// Matrix Addition
const matrixAdd = math.add(mA, mB);
// Result [ [2, 1], [5, 2], [8, 3] ]
If two matrices have the same dimension, we can subtract them:
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const mA = math.matrix([[1, 2], [3, 4], [5, 6]]);
const mB = math.matrix([[1,-1], [2,-2], [3,-3]]);
// Matrix Subtraction
const matrixSub = math.subtract(mA, mB);
// Result [ [0, 3], [1, 6], [2, 9] ]
To add or subtract matrices, they must have the same dimension.
While numbers in rows and columns are called Matrices, single numbers are called Scalars.
It is easy to multiply a matrix with a scalar. Just multiply each number in the matrix with the scalar:
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x 2 = |
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const mA = math.matrix([[1, 2], [3, 4], [5, 6]]);
// Matrix Multiplication
const matrixMult = math.multiply(2, mA);
// Result [ [2, 4], [6, 8], [10, 12] ]
const mA = math.matrix([[0, 2], [4, 6], [8, 10]]);
// Matrix Division
const matrixDiv = math.divide(mA, 2);
// Result [ [0, 1], [2, 3], [4, 5] ]
To transpose a matrix, means to replace rows with columns.
When you swap rows and columns, you rotate the matrix around it's diagonal.
A = |
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AT = |
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Multiplying matrices is more difficult.
We can only multiply two matrices if the number of rows in matrix A is the same as the number of columns in matrix B.
Then, we need to compile a "dot product":
We need to multiply the numbers in each row of A with the numbers in each column of B, and then add the products:
const mA = math.matrix([[1, 2, 3]]);
const mB = math.matrix([[1, 2, 3], [1, 2, 3], [1, 2, 3]]);
// Matrix Multiplication
const matrixMult = math.multiply(mA, mB);
// Result [ [6, 12, 18] ]
Explained:
A | B | C | C | |||||||||||||||||||||
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If you know how to multiply matrices, you can solve many complex equations.
You sell roses.
What was the value of all the sales?
$3 | $4 | $2 | |
Mon | 120 | 80 | 60 |
Tue | 90 | 70 | 40 |
Wed | 60 | 40 | 20 |
A | B | C | C | |||||||||||||||||||
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const mA = math.matrix([[3, 4, 2]]);
const mB = math.matrix([[120, 90, 60], [80, 70, 40], [60, 40, 20]);
// Matrix Multiplication
const matrixMult = math.multiply(mA, mB);
// Result [ [800, 630, 380] ]
Explained:
A | B | C | C | |||||||||||||||||||||
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With AI, you need to know how to factorize a matrix.
Matrix factorization is a key tool in linear algebra, especially in Linear Least Squares.