A matrix and its transpose have the same set of eigenvalues/other version: AA and ATA^T have the same spectrum

Let σ(A) be the set of all eigenvalues of A. Show that σ(A)=σ(AT) where AT is the transpose matrix of A.

Answer

The matrix (AλI)T is the same as the matrix (ATλI), since the identity matrix is symmetric.

Thus:

det(ATλI)=det((AλI)T)=det(AλI)

From this it is obvious that the eigenvalues are the same for both A and AT.

Attribution
Source : Link , Question Author : Zizo , Answer Author : Invisible

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