TITLE: Properties shared by similar and unitary similar matrices.
QUESTION [10 upvotes]: We know that matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A=PBP^{-1}$ and they are unitarily similar if $P$ is unitary ($PP^*=P^*P=I$). I want to know : What are the properties of the matrix that are preserved by these transformations ?.  I'm a little confident of a few properties, but there are some I'm not sure of. 
What I need is :  1) Answers to those places which I not sure of and corrections for existing answers.2) Interpretation for the differences in similarity and unitary similarity.3) Other properties I might have missed out here. Thanks a lot.
\begin{array}{ccc}&Property&Similarity & Unitary \ Similarity  
\\\hline 1.&Characteristic \ polynomial&Yes&Yes
\\2.&Eigenvalues&Yes&Yes
\\3.&Geometric \ multiplicity \ of \ eigenvalues&Yes&Yes
\\4.&Elementary \ divisors \ and \ Rational \ canonical \ form&Yes&Yes
\\5.&Positive\ definiteness&Yes&Yes
\\6.& Matrix \ 2-Norm&Yes&Yes
\\7.&Symmetry&No&Yes
\\8.&SkewSymmetry&No&Yes
\\9.&Orthogonality&No&Yes
\\10.&Frobenius\ Norm&No&No
\\11.&1-norm  \ and  \ \infty- norm&No&No
\\12.&Unitary \ Diagnalizability& No& Not\ sure
\\13.&Nilpotency&Not \ sure& Not \ sure
\\14.&Solution \ to \ Ax=b&Not \ sure& Not \ sure
\\15.&Minimal \ polynomial&Not \ sure&Not \ sure
\\16.&Diagonalizability&Not \ sure& Not \ sure
\\17.&Rank &Not\ sure& Not\ sure
\\18.&Singular\ Values&Not \ Sure&Not \ Sure
\\19.&Condition\ Number&Not \ Sure&Not \ Sure
\end{array}

REPLY [2 votes]: Note that unitary similarity implies similarity, so properties holding for all similar matrices hold for all unitarily similar matrices.
Nilpotence is preserved for both as we have (by induction on $k$)
$$A^k=0 \implies (PBP^{-1})^k=PB^kP^{-1}=0\implies B^k=0$$
Solution sets are not preserved for similar matrices as $PAP^{-1}x\neq Ax$ for the following invertible $A$ and $P$ and vector $b$ (one of many counterexamples):
\begin{align}A=\begin{bmatrix}1&2\\0&1\end{bmatrix}&&P=\begin{bmatrix}1&1\\0&1\end{bmatrix}&&x=\begin{bmatrix}2\\1\end{bmatrix}\end{align}
Diagonalizability is preserved for both as $D=QAQ^{-1}$ for some invertible $Q$ implies $D=(QP)B(QP)^{-1}$ and $QP$ is invertible.
Minimal Polynomials are preserved for both as a result of a heap of theory.
Rank is preserved for both by uniqueness of reduced row echelon form and invariance thereof under similarity.
Unitary Diagonalizability is preserved for unitarily similar matrices as $D=QAQ^*=(QP)B(QP)^*$ and $QP$ is unitary.