Orthogonal matrix whose submatrix has special properties

Thread starter julie94
Start date Oct 20, 2009
Tags

Matrix Orthogonal Properties

In summary, an orthogonal matrix is a square matrix with orthogonal rows and columns, meaning their dot product is zero. It is represented by the letter Q and its transposition is written as QT. The submatrix of an orthogonal matrix has the same properties and dimensions as the original matrix. All rows and columns in an orthogonal matrix must be orthogonal, including the submatrix. Applications of orthogonal matrices with special submatrix properties include signal processing, image compression, data encryption, and solving systems of equations in linear algebra and geometry.

Oct 20, 2009

julie94

Dear Forumers.

I am working on the following problem.

Let matrix P=( A B ) where A and B are matrices. Let P be an n*n orthogonal matrix.

Show that A'A is an idempotent matrix.

I do not know where to start. Thanks in advance for the help.

Physics news on Phys.org

Oct 20, 2009

julie94

I can write
PP'=(A B)(A B)'
=(AB'+AA' BB' +BA')

and I can write
PP'=In

FAQ: Orthogonal matrix whose submatrix has special properties

What is an orthogonal matrix?

An orthogonal matrix is a square matrix in which all the columns and rows are orthogonal, meaning they are perpendicular to each other. This means that the dot product of any two columns or rows is equal to zero.

How is an orthogonal matrix represented?

An orthogonal matrix is typically represented using the letter Q, and is often written as QT when transposed.

What are the special properties of a submatrix in an orthogonal matrix?

A submatrix in an orthogonal matrix has the same properties as the original matrix, meaning it is also orthogonal. Additionally, the submatrix will have the same dimension as the original matrix.

Can a submatrix in an orthogonal matrix have rows or columns that are not orthogonal?

No, by definition, all rows and columns in an orthogonal matrix, including the submatrix, must be orthogonal.

What are some applications of orthogonal matrices with special submatrix properties?

Orthogonal matrices with special submatrix properties have many applications in fields such as signal processing, image compression, and data encryption. They are also commonly used in linear algebra and geometry to simplify calculations and solve systems of equations.