# Are Orthogonal Matrices Compact?

• blaster
In summary, an orthogonal matrix is a square matrix with mutually perpendicular rows and columns. Its compactness is achieved by having a diagonal matrix with only 1s and -1s. These matrices have various applications in mathematics, science, and computer graphics. They are different from other matrices due to their unique properties and can be calculated using the Gram-Schmidt process or QR decomposition method.
blaster
How can you prove that the set of orthogonal matrices are compact? I know why they are bounded but do not know why they are closed.

What is the topology?

(By the way- closed and bounded does not necessarily imply compact.)

how about defining them by an equation?

and of course closed and bounded does imply compact in a euclidean space such as matrix space. as hurkyl knows very well - he is just trying to scare you.

(in a general metric space you need "complete and totally bounded")

## 1. What is an orthogonal matrix?

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

## 2. How is compactness related to orthogonal matrices?

Compactness refers to the property of a matrix where its elements are arranged in a way that minimizes the space it occupies. In the case of orthogonal matrices, compactness is achieved by having a diagonal matrix with only 1s and -1s, resulting in a more efficient use of space.

## 3. What are the applications of orthogonal matrices?

Orthogonal matrices have several applications in mathematics and science. They are commonly used in linear algebra, signal processing, and statistics. They also play a significant role in computer graphics, where they are used to rotate and transform objects in 3D space.

## 4. How are orthogonal matrices different from other types of matrices?

Unlike other types of matrices, orthogonal matrices have some unique properties. They are square matrices that are invertible, meaning they have a unique inverse matrix. They also preserve the length of vectors and angles between vectors, making them useful for geometric transformations.

## 5. How are orthogonal matrices calculated?

To calculate an orthogonal matrix, one can use the Gram-Schmidt process to transform a set of linearly independent vectors into a set of orthogonal vectors. Alternatively, one can also use the QR decomposition method, where a matrix is decomposed into an orthogonal matrix and an upper triangular matrix.

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