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Math Help - orthogonal matrix

  1. #1
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    Cool orthogonal matrix

    hi,

    im not sure this question is supposed to be under what topic but here is my question:

    given taht A is a sq matrix

    (A^t)A = I iff (Ai)(Ai) =1 for all i.

    also, what does it mean by an orthonormal basis being a basis consisting of mutually orthogonal unit vectors? does it mean that if A is a sq matrix, then the column vectors in A are orthogonal to each other?

    thanks!
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    hi,

    im not sure this question is supposed to be under what topic but here is my question:

    given taht A is a sq matrix

    (A^t)A = I iff (Ai)(Ai) =1 for all i.
    Well, I don't think that this is quite true, maybe you wanted to write something like

    A^{\top}A=I \Leftrightarrow A_i A_j = \delta_{ij}, \forall i,j

    also, what does it mean by an orthonormal basis being a basis consisting of mutually orthogonal unit vectors?
    Just what it says: that it is a basis and, in addition, that these vectors are normalized and mutually orthogonal. Of course, a set of normalized, mutually orthogonal vectors is always linearly independent, but it need not be a basis (i.e. it may not span the entire vector space).

    does it mean that if A is a sq matrix, then the column vectors in A are orthogonal to each other?
    That does not follow at all. How could it? For example,

    A:= \begin{pmatrix}1&1\\1&2\end{pmatrix}
    is clearly a square matrix, but its column vectors are certainly not orthogonal to each other.

    Note that from A^{\top}A=I it follows that AA^{\top}=I. This is because A^{\top}A=I means that A^{\top} is the inverse of A, and inverses commute...
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  3. #3
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    i was thinking if the orthonormal basis is expressed as a matrix, then wont it mean that the column vectors are orthogonal?

    thanks for explaining it to me.
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    i was thinking if the orthonormal basis is expressed as a matrix, then wont it mean that the column vectors are orthogonal?
    To my ears, "expressing an orthonormal basis as a matrix" sounds a little funny. If you mean to say by this turn of phrase that the matrix that maps an orthonormal basis (such as the standard basis) to another orthonormal basis, then, yes, the column vectors of that matrix form an orthonormal basis, and, by definition, that matrix is orthogonal.

    This means that the row vectors of that matrix form an orthonormal basis as well.
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