Elementary Operations on Matrices

In this module, you will be introduced to some elementary operation on scalars, vectors and matrices that are key linear algebra.

Today's learning objectives

  • Understand matrix transpose
  • Understand special matrices - diagonal, identity, and symmetric, orthagonal.
  • Learn how to compute the determinant of a matrix.

Transpose of a 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 from the top left and to the bottom right.

We denote the transpose of a matrix A as AT.


Please watch this brief video to understand how to obtain the transpose of a matrix:

Special matrices

There are certain ‘special matrices' that you will encounter several times during your study. These matrices are special because of certain inherent properties they posses. In this course, you will study about:

  1. Identity matrix
  2. Diagonal matrix
  3. Symmetric matrix
  4. orthagonal matrix

Identity Matrix

An identity matrix is a matrix that does not change any vector when we multiply that vector by that matrix. In simpler terms, an identity matrix is a matrix with only 1's along the main diagonal.


Diagonal Matrix

A diagonal matrix is a matrix in which all off-diagonal entries are zero. In mathematical terms, a matrix A is said to be diagonal is Ai,j = 0 ; for all i =/= j. Such matrices consist mostly of zeros and have non-zero entries only along the main diagonal. Thus, an identity matrix is by default, also a diagonal matrix.


Symmetric Matrix

A symmetric matrix is any matrix that is equal to its own transpose (A = AT). Again, an identity matrix is by default, also a symmetric matrix.

Orthogonal Matrix

Two vectors x and y are said to be orthagonal to each other is xTy = 0. Along these lines, a matrix A is said to be orthogonal if it's rows and columns are mutually orthagonal.

Determinant of a Matrix

The determinant of a matrix, denoted by det(A) or |A| is a function which maps a given matrix to a scalar. To compute the determinant, in the case of a 2 X 2 matrix, it is given by:


In the case of a 3X3 matrix, it is given by:


At this moment, it is sufficient to understand how to compute a determinant of a matrix. As you progress in your study, you will slowly start to Understand more about the properties of a determinant and what makes it so useful to compute!

Assignment

a) A correlation matrix is _____ (pick the right answer(s))

  • Always a diagonal matrix
  • Always a symmetric matrix
  • Always an identity matrix
  • Always an orthagonal matrix
  • Possibly a diagonal matrix
  • Possibly a symmetric matrix
  • Possibly an identity matrix
  • Possibly an orthagonal matrix
  • Never a diagonal matrix
  • Never a symmetric matrix
  • Never an identity matrix
  • Never an orthagonal matrix

hint: create a correlation matrix in python using numpy and check which of the conditions hold true

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