What is an antidiagonal?

Anonymous

∙ 9y ago

Updated: 8/21/2019

An antidiagonal is a diagonal of a matrix which leads from top-right to bottom-left.

Wiki User

∙ 9y ago

What else can I help you with?

How do you show that a square matrix A is similar to its transpose?

First we will handle the diagonalizable case.Assume A is diagonalizable, A=VDV-1.Thus AT=(V-1)TDVT,and D= VT AT(V-1)T.Finally we have that A= VVT AT(V-1)TV-1, hence A is similar to ATwith matrix VVT.If A is not diagonalizable, then we must consider its Jordan canonical form,A=VJV-1, where J is block diagonal with Jordan blocks along the diagonal.Recall that a Jordan block of size m with eigenvalue at L is a mxm matrix having L along the diagonal and ones along the superdiagonal.A Jordan block is similar to its transpose via the permutation that has ones along the antidiagonal, and zeros elsewhere.With this in mind we proceed as in the diagonalizable case,AT=(V-1)TJTVT.There exists a block diagonal permutation matrix P such thatJT=PJPT, thus J=PTVT AT(V-1)TP.Finally we have that A= VPTVT AT(V-1)TPV-1, hence A is similar to ATwith matrix VPTVT.Q.E.D.

What is an antidiagonal?

What else can I help you with?

How do you show that a square matrix A is similar to its transpose?

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