Sets of vectors are orthogonal or orthonormal. There is no such thing as an orthonormal matrix. An orthogonal matrix is a square matrix whose columns (or rows) form an orthonormal basis. The terminology is unfortunate, but it is what it is.

So, basically, orthogonal matrix is just a combination of one-dimensional reflectors and rotations written in appropriately chosen orthonormal basis (the coordinate system you're used to, but possibly rotated). Fun fact: All orthogonal matrices (even rotations) of order n n can be presented as compositions of at most n n reflectors.

Orthogonal matrices are invertible square matrices, so their singular values are their eigenvalues. Their eigenvalues are complex numbers whose norm (i.e. absolute value) is 1 1, or in other words, they're all on the circle of unit radius centered at 0 0 in the complex plane.

Sep 1, 2015 · I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal matrix, I mean an n × n n × n matrix with orthonormal columns.

Sep 23, 2011 · P.S. Householder matrices (elementary reflectors) are a typical example of matrices that are symmetric, orthogonal, and involutory.

The original question was asking about a matrix H and a matrix A, so presumably we are talking about the operator norm. The selected answer doesn't parse with the definitions of A and H stated by the OP -- if A is a matrix or more generally an operator, (A,A) is not defined (unless you have actually defined an inner product on the space of ...

Well, if the columns are orthonormal (i.e. norm 1), then the matrix is orthogonal, and has many beautiful properties. If not, see Name for matrices with orthogonal (not necessarily orthonormal) rows. I suppose the right way to think about it is that this matrix maps the standard basis vectors to an orthogonal basis.

You can construct orthogonal and symmetric matrices using a nice parametrization from Sanyal [1] and Mortari [2]. We want a matrix R R both orthogonal and symmetric, i.e.

The second statement should say that the determinant of an orthogonal matrix is ± 1 and not the eigenvalues themselves. R is an orthogonal matrix, but its eigenvalues are e ± i.

May 2, 2015 · The term "orthogonal matrix" probably comes from the fact that such a transformation preserves orthogonality of vectors (but note that this property does not completely define the orthogonal transformations; you additionally need that the length is not changed either; that is, an orthonormal basis is mapped to another orthonormal basis).