- #1
DmytriE
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Can someone confirm or refute my thinking regarding the diagonalizability of an orthogonal matrix and whether it's symmetrical?
A = [b1, b2, ..., bn] | H = Span {b1, b2, ..., bn}. Based on the definition of the span, we can conclude that all of vectors within A are linearly independent. Furthermore, we can then conclude that Rank(A) = n.
If Rank(A) = n then none of the vectors in A can be made as a linear combination of the other n-1 vectors. Since A can be row-reduced to an identity matrix and the transpose of the identity matrix is itself. Can it be concluded that the original matrix A is also symmetrical (AT = A)?
A = [b1, b2, ..., bn] | H = Span {b1, b2, ..., bn}. Based on the definition of the span, we can conclude that all of vectors within A are linearly independent. Furthermore, we can then conclude that Rank(A) = n.
If Rank(A) = n then none of the vectors in A can be made as a linear combination of the other n-1 vectors. Since A can be row-reduced to an identity matrix and the transpose of the identity matrix is itself. Can it be concluded that the original matrix A is also symmetrical (AT = A)?