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}
Best Answer
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.