Introduction to Basic Foundation 1 - Linear Algebra and Calculus

General notations

We note a vector with entries, where is the entry:

We note a matrix with rows and columns, where is the entry located in the row and column:

Remark: the vector defined above can be viewed as a matrix and is more particularly called a column-vector.

The identity matrix is a square matrix with ones in its diagonal and zero everywhere else:

Remark: For all matrices , we have .

A diagonal matrix is a square matrix with nonzero values in its diagonal and zero everywhere else:

Remark: we also note as .

There are two types of vector-vector products:

Inner product: for , we have:
outer product: for and , we have:
Matrix-vector multiplication The product of matrix and vector is a vector of size , such that:
Matrix-matrix multiplication The product of matrices and is a matrix of size , such that:
where are the vector rows and are the vector columns of and are the entries of .
Transpose The transpose of a matrix , noted , is such that its entries are flipped:

Remark: for matrices , we have .

Inverse The inverse of an invertible square matrix is noted and is the only matrix such that:

Remark: not all square matrices are invertible. Also, for matrices , we have .

Trace The trace of a square matrix , noted , is the sum of its diagonal entries:

Remark: for matrices , we have and .

Determinant The determinant of a square matrix , noted or , is expressed recursively in terms of which is the matrix without its row and column, as follows: