Wouldn't that be information you should include in your first post, as either given/known data or relevant equations?Dustinsfl said:The columns most for an orthogonal set.
Mark44 said:Wouldn't that be information you should include in your first post, as either given/known data or relevant equations?
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 zero, and the norm (length) of each column or row is equal to one. Orthogonal matrices are important in linear algebra because they preserve the length and angle of vectors, making them useful for transformations and solving systems of equations.
The transpose of a matrix is a new matrix where the rows and columns are flipped. This means that the first row becomes the first column, the second row becomes the second column, and so on. More generally, the element in the i-th row and j-th column of the original matrix becomes the element in the j-th row and i-th column of the transposed matrix. The transpose of a matrix is denoted by a superscript "T" next to the matrix (e.g. AT).
To prove that the transpose of an orthogonal matrix is orthogonal, we need to show that the dot product of any two columns or rows is equal to zero, and the norm of each column or row is equal to one. This can be done by using the definition of the transpose and the properties of dot products and norms.
Proving that an orthogonal matrix transpose is orthogonal is important because it confirms that the matrix preserves the length and angle of vectors. This is useful in many applications, such as solving systems of equations, performing rotations and reflections, and in computer graphics and machine learning.
No, an orthogonal matrix transpose can only be a square matrix. This is because the transpose operation only flips the rows and columns of a matrix, so the resulting matrix will always have the same number of rows and columns as the original matrix. Since orthogonal matrices are always square matrices, their transpose will also be a square matrix.