Quantum Talk

In multivariable calculus, there are three important and independent operators that can be combined in a variety of ways to give other operators. The three main operators as discussed in the previous post are Gradient, Divergence and Curl. These operators can be combined in ways that give other operators of which the best example is Laplacian. Laplacian is basically the divergence of gradient of a scalar field. Laplacian is defined for vector valued functions as well and in this case, each components is treated as a scalar and taken Laplacian of. The end result is again a vector. This post talks about them in detail.

One of the most important concepts in modern physics is Eigenvalues and Eigenvectors of a matrix. Eigenvectors are also termed as the invariant vectors i.e. they are not changed under matrix transformation other than their length. This property of invariance makes it very useful for understanding Eintein's theory of relativity and quantum mechanics.

Another important term used in matrix operation is its rank. The rank of a matrix is the number of independent Eigenvectors it has. It is also defined as the high order non-zero minor of the matrix.

Today, I attended my first lecture on Matrices (Mathematics) and here are the basic properties of orthogonal matrices that we learnt.

Definition of Orthogonal Matrix
A matrix whose inverse is equal to its transpose are termed as Orthogonal matrices.

Special Properties of Orthogonal Matrices
1. The determinant of an orthogonal matrix is either 1 or -1.
2. The product of two orthogonal matrices is also orthogonal.
3. The inverse of an orthogonal matrix is also orthogonal.

The following images provide the basic proofs of these properties which uses properties of matrix algebra (associativity) and transpose of matrices. Hope you find that informative and useful.