Determinant of an orthogonal matrix

In summary, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. The determinant of an orthogonal matrix is always either 1 or -1, and it has properties such as having a transpose equal to its inverse. In linear algebra, it is commonly used for rotations, reflections, orthonormalization, solving systems of equations, and computing eigenvalues and eigenvectors. It must have the same number of rows and columns to be orthogonal.
  • #1
squenshl
479
4
How is it the determinant of an orthogonal matrix is [tex]\pm1[/tex].
Is it:
Suppose Q is an orthogonal matrix [tex]\Rightarrow [/tex] 1 = det(I) = det(QTQ)
= det(QT)det(Q) = ((det(Q))2
and if so, what is it for -1.
Thanks.
 
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  • #2
Never mind, got it.
 
  • #3
Maybe you can share your new knowledge with the forum so that when other users see the question they can also see the answer!
 
  • #4
Sweet.
Suppose Q is orthogonal, then ATA = In
[tex]\Rightarrow[/tex] det(A)det(ATA) = det(A)det(A) = (det(A))2 = 1
Which implies that det(A) = [tex]\pm\sqrt{1}[/tex] = [tex]\pm1[/tex]
 

FAQ: Determinant of an orthogonal matrix

What is an orthogonal matrix?

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. This means that the dot product of any two columns (or rows) is equal to 0, and the length of each column (or row) is equal to 1.

How do you calculate the determinant of an orthogonal matrix?

The determinant of an orthogonal matrix is always either 1 or -1. This is because the product of all the diagonal elements (also known as the main diagonal) is equal to the determinant, and the diagonal elements of an orthogonal matrix are all either 1 or -1.

What are some properties of an orthogonal matrix?

Some key properties of an orthogonal matrix include:

  • The transpose of an orthogonal matrix is equal to its inverse.
  • The determinant is always either 1 or -1.
  • The columns (and rows) are orthogonal unit vectors.
  • The product of two orthogonal matrices is also an orthogonal matrix.

How is an orthogonal matrix used in linear algebra?

An orthogonal matrix is commonly used in linear algebra for various purposes, such as:

  • Performing rotations and reflections in n-dimensional space.
  • Orthonormalizing a set of vectors.
  • Solving systems of equations.
  • Computing eigenvalues and eigenvectors.

Can a non-square matrix be orthogonal?

No, a non-square matrix cannot be orthogonal. In order for a matrix to be orthogonal, it must have the same number of rows and columns.

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