An Idea On Spectral Decomposition

What Is An Orthogonal Matrix?

A square matrix A would be an orthogonal matrix only if its transpose is equal to its inverse. AT = A-1, where AT is the transpose of the square matrix A and A-1 is its inverse. 

Also, with this definition, we will arrive at another definition of an orthogonal matrix.

AT = A-1

Multiplying both sides by A, we get

AAT = AA-1

Now, AA-1 = I, where I is called the identity matrix having the same order as that of A

So, AAT = I

Also, we can prove that ATA = I on a similar basis.

Thereby, we arrive at two definitions of an orthogonal matrix:

1. AT = A-1
2. AAT = ATA = I

Properties Of Orthogonal Matrix

Some of the properties of an orthogonal matrix are:

1. If all the eigenvalues of an orthogonal matrix are real, then the eigenvalues would always be 1.
2. An orthogonal matrix should be symmetric.
3. If A is n n symmetrical matrix such that A2 = I, then A is orthogonal.
4. Product of two orthogonal matrices is also orthogonal.
5. The determinant of an orthogonal matrix is always 1.

What are Eigenvalues?

Developed by English mathematician Arthur Cayley, eigenvalues are a type of scalar quantities which is associated with linear equation systems (matrices). Simply put, eigenvalue of square matrix A is scalar  , such that A = λν . Here,  is an eigenvector for  . Eigenvalues can be equal to zero.

What are Eigenvalues used for?

It is used in linear equations in order to simplify them. Eigenvalues help to determine data variation in a specific direction. eigenvalues are a type of scalar quantity which is associated with linear equation systems (matrices).

What Is Spectral Decomposition Of A Matrix?

The term ‘spectral decomposition’ was coined by German mathematician David Hibert in about 1905. On the basis of the spectral theorem, a matrix when broken down or decomposed as a symmetrical matrix, then this type of decomposition is referred to as spectral decomposition.

Basically, spectral decomposition rearranges a matrix in terms of eigenvalues and eigenvectors. This results in obtaining the original matrix by the summation of these terms. With respect to eigenvalues, a spectral decomposition is not unique. It was initially created for symmetric matrices. 

Spectral decomposition is useful for looking into the various properties of a given matrix. Some complex operations such as determining the power of a given matrix, can be feasibly performed with the help of eigenvalue spectral decomposition.

Eigenvalue Spectrum

The spectrum of a matrix is represented as its eigenvalue sets. Simply put, as eigenvalues operate in one-dimensional space, the spectrum represents this scalar system which is not invertible. 

Conclusion

In this article, we discussed an orthogonal matrix. A square matrix A would be an orthogonal matrix only if its transpose is equal to its inverse. AT = A-1, where AT is the transpose of the square matrix A and A-1 is its inverse. Moving on, we discussed eigenvalues along with its uses. Eigenvalues are a type of scalar quantity which is associated with linear equation systems (matrices). We also discussed spectral decomposition of a matrix, which rearranges a matrix in terms of eigenvalues and eigenvectors. This results in obtaining the original matrix by the summation of these terms.