2. Linear Algebra (Matrix operations)

2. Matrix operations

The following re the common operations are performed on matrix in ML

1.    1.  Transpose

2.     2. Addition of Matrix

3.     3. Subtraction of Matrix

4.     4. Multiplication of matrix

One important operation on matrices is the transpose. The transpose of a matrix is the mirror image of the matrix across a diagonal line, called the main diagonal, running down and to the right, starting from its upper left corner. We can represent as (A^T)i,j = Aj,i

Vectors can be thought of as matrices that contain only one column. The transpose of a vector is therefore a matrix with only one row. A vector by writing out its elements in the text inline as a row matrix, then using the transpose operator to turn it into a standard column vector,

 e.g., x = [x1, x2, x3 ]^T

2 &3 . Addition  & Subtraction of Matrix

We can add matrices to each other, as long as they have the same shape, just by adding their corresponding elements.

Example C = A + B   where Ci,j = Ai,j + Bi,j .

We can also add a scalar to a matrix or multiply a matrix by a scalar, just by performing that operation on each element of a matrix.

Exaple  D = a * B + c  where Di,j = a * Bi,j + c

4. Multiplying Matrices and Vectors

One of the most important operations involving matrices is multiplication of two matrices. The matrix product of matrices A and B is a third matrix C. In order for this product to be defined, A must have the same number of columns as B has rows. If A is of shape m × n and B is of shape n × p, then C is of shape m × p. We can write the matrix product just by placing two or more matrices together, e.g. C= AB

The product operation is defined by C_i,j =  ∑ A_i,kB_k,j

Matrix product operations have many useful properties that make mathematical analysis of matrices more convenient.

 For example, matrix multiplication is distributive:

A(B + C) = AB + AC

 It is also associative:

A(BC) = (AB)C

Matrix multiplication is not commutative (the condition AB = BA does not always hold), unlike scalar multiplication. However, the dot product between two vectors is commutative: x^Ty = y^Tx. 

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