Trace Properties and Applications

Trace = sum of diagonal elements = sum of eigenvalues

Property	Formula
tr(A + B)	tr(A) + tr(B)
tr(kA)	k * tr(A)
tr(AB)	tr(BA) -- cyclic property
tr( $A^T$ )	tr(A)
tr( $I_n$ )	n
tr(A)	$lambda_1$ + $lambda_2$ + ... + $lambda_n$

Connection to characteristic equation (2x2): Characteristic equation: $lambda^2$ - (tr A)lambda + det(A) = 0

Connection to characteristic equation (3x3): $lambda^3$ - (tr A) $lambda^2$ + (sum of 2x2 principal minors)lambda - det(A) = 0

This means for 2x2 matrices, knowing trace and determinant completely determines the characteristic equation, eigenvalues, and by Cayley-Hamilton, all powers of A.