Linear algebra matrix

Linear Algebra

Scalar scalar

        Individual numbers, natural numbers, integers, real numbers,,

        Lowercase italics indicates

Vector vector

• A set of one-dimensional array
• A number of ordered, generally define the longitudinal amount.
• However, writing is not convenient, the use of multiple vectors transposed Writing
• Vector generally indicated in bold lowercase variable names, such as x
• A set of vector elements, define the set S = {1,3,6}, and then written as xs

• Elements of the vector is represented by italics with subscript is, as a first vector x elements x1, x2 of two elements
• A set of vector elements, define the set S = {1,3,6}, and then written as xs

Matrix matrix

        Two-dimensional array

Generally represents a matrix, such as bold uppercase A variable name

• Usually bold uppercase variable name represented by a matrix, such as A

• A _{i, j} represents the matrix row i, column j element
• f ( A ) _{i, j} denotes role in the function f A i-th row of the output matrix elements on the j-th column.
• In the data row for general

Tensor tensor

• More than two-dimensional array
• Shape refers to the tensor dimension
• Shape (2,5) represents a matrix of rows and five columns
• For example, shape is (2,3,4) tensor
• Tensorflow: tensor stream
• Scalar, vector, matrix are also special tensor

Transpose

• Vector conversion ranks
• Diagonal axis mirror
• Matrix transpose, satisfying
• 
• Vector can be regarded as only one of the matrix, which can be regarded as only one line permutation matrix, a vector is defined as:

• Only a scalar element, is equal to the transpose of its own,

Matrix addition

Matrix subtraction

 

Matrix Multiplication

The final results of the new matrix of columns in the row for the A's Xb

Matrix multiplication formula

        

Multiplication matrix element corresponding to element wise product

        Shape one kind of a product using the same

Matrix dot product dot product

 

For the shape of one-dimensional array for array element number

        The vector dot product is a scalar and a value

 

        Exemplary dot product of two vectors

x = [1,2,3]^T

y = [7,9,11]^T

x.y = x^Ty = [1,2,3].[7,9,11]^T = 58

Matrix

Matrix structure is simple: all elements along the main diagonal is 1, while the other elements are zero position

• Properties: arbitrary vectors, matrices and matrix multiplication, will not change.
• Consistent with the ranks of the identity matrix
• The n-dimensional vector generally remains constant matrix referred to as
• formal:

Linear Equations

        Matrix is ​​an important tool for the solution of linear equations

        Linear equations written in another way

Inverse matrix

• The result is multiplied by a matrix destination matrix is ​​the identity matrix, the target torque reversible,
• Inverse matrix and the matrix is ​​a matrix of target
• Referred to as the matrix inverse matrix satisfies the following conditions:

  

• given,

  

• We can solve the vector through the following steps