Wednesday, August 7, 2019

Eigenvalues of a Matrix and its Transpose are the Same



https://yutsumura.com/eigenvalues-of-a-matrix-and-its-transpose-are-the-same/

Recall that the eigenvalues of a matrix are roots of its characteristic polynomial.
Hence if the matrices A and AT have the same characteristic polynomial, then they have the same eigenvalues.
So we show that the characteristic polynomial pA(t)=det(AtI) of A is the same as the characteristic polynomial pAT(t)=det(ATtI) of the transpose AT.
We have
pAT(t)=det(ATtI)=det(ATtIT)since IT=I=det((AtI)T)=det(AtI)since det(BT)=det(B) for any square matrix B=pA(t).

Therefore we obtain pAT(t)=pA(t), and we conclude that the eigenvalues of A and AT are the same.

Remark: Algebraic Multiplicities of Eigenvalues

Remark that since the characteristic polynomials of A and the transpose AT are the same, it furthermore yields that the algebraic multiplicities of eigenvalues of A and AT are the same.

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