Transpose of orthogonal matrix

In summary, an orthogonal matrix is one where Q^TQ=I, but not necessarily QQ^T=I. Therefore, the inverse of Q can be represented as Q^T. The question of whether QQ^T=I is 100% true is still open, but in cases where Q has more rows than columns, QTQ still equals I.
  • #1
td21
Gold Member
177
8

Homework Statement



Orthogonal matrix means [itex]Q^{T}Q=I[/itex], but not necessary [itex]QQ^{T}=I[/itex], so why can we say the inverse of Q is [itex]Q^{T}[/itex]?

Homework Equations





The Attempt at a Solution


the attempt is actually in my question. It's something i don't understand when doing revision.
 
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  • #2
addition:
is [itex]
QQ^{T}=I
[/itex]
100% true?If it is, my problem is solved.
 
  • #3
td21 said:
addition:
is [itex]
QQ^{T}=I
[/itex]
100% true?If it is, my problem is solved.

Sure. Q^TQ=I implies QQ^T=I.
 
  • #4
An interesting variation is when Q has more rows than columns. In this case, QTQ still equals I but QQT doesn't.
 

Related to Transpose of orthogonal matrix

1. What is an orthogonal matrix?

An orthogonal matrix is a square matrix where each column and row is orthogonal to each other. This means that the dot product of any two columns or rows is equal to 0, making the matrix's transpose equal to its inverse.

2. What is the transpose of an orthogonal matrix?

The transpose of an orthogonal matrix is simply the matrix itself, as the transpose of an orthogonal matrix is equal to its inverse. This is because the dot product of any two columns or rows is 0, making the matrix's transpose equal to its inverse.

3. How is the transpose of an orthogonal matrix calculated?

The transpose of an orthogonal matrix can be calculated by simply switching the rows and columns of the matrix. This is because the dot product of any two columns or rows is 0, making the transpose equal to the inverse of the matrix.

4. What are the properties of an orthogonal matrix?

An orthogonal matrix has several properties, including: all columns and rows are orthonormal, the determinant is either 1 or -1, the inverse is equal to the transpose, and the transpose of the transpose is equal to the original matrix.

5. How is an orthogonal matrix used in mathematics and science?

Orthogonal matrices are widely used in mathematics and science, particularly in linear algebra and geometry. They are used for solving systems of equations, finding eigenvalues and eigenvectors, and transforming vectors and coordinates in geometric transformations. They also have applications in fields such as computer graphics, signal processing, and quantum mechanics.

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