Properties of ATA transformation. Is it true a proposition for 3×3 real matrices A that if for a some nonzero vector v we have Av≠v and at the same time (ATA)v=v then for any vector u from R3 we have (ATA)u=u (i.e. A is orthogonal) ? Do we need assumption about full rank of the matrix A for this goal? Can we extend the proposition for bigger dimensions?

Question

Properties of ATA transformation.  Is it true a proposition for 3×3 real matrices A that if for a some nonzero vector v we have Av≠v and at the same time (ATA)v=v then for any vector u from R3 we have (ATA)u=u (i.e. A is orthogonal) ?   Do we need assumption about full rank of the matrix A for this goal?   Can we extend the proposition for bigger dimensions?

Iacopelli5co · Accepted Answer

Your claim fails even if A is of full rank: for example, consider  A=[−100020001] and v=[100]  Then ATA=A2 and you can easily find u such that ATAu≠u.

Justice Jacobson · Accepted Answer

This is wrong.  Take any orthogonal matrix M∈R2×2, and extend it into a 3×3 matrix using a column of ones and a row of zeroes:  [[M]11001]  The result is not an orthogonal matrix, but it has full rank and even determinant 1, and most of the time v=(100) will satisfy your requirements.

Properties of A^{T}A transformation. Is it true a proposition for 3 \times 3 real ma

Answered question

Answer & Explanation

New Questions in Integral Calculus