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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}

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  • $\begingroup$ Concerning the question in the title (not much related to the content of the post), note that similarity of matrices is an equivalence relation. $\endgroup$ – Algebraic Pavel Apr 16 '15 at 9:41
  • $\begingroup$ You wrote that similarity does not preserve symmetry (or rather conjugate symmetry in case of complex matrices) but preserves positive definiteness. What does this mean in particular when matrices are complex? Also, why similarity preserves 2-norm? Why unitary similarity does not preserve the Frobenius norm? What is the "condition number" (there are many condition numbers)? $\endgroup$ – Algebraic Pavel Apr 16 '15 at 9:48
  • $\begingroup$ @AlgebraicPavel Regarding positive definiteness, what I meant was the condition that "hermitian matrices all of whose eigenvalues are positive " and so that's a mistake since I don't know if "positive definiteness" is even defined for non-hermitian matrices. Regarding 2-norm, I meant the largest eigenvalue. Frobenius norm is the square root of sum of squares of elements of the matrix. I think there must be counterexample to prove that this is not preserved. By Condition number I meant the ratio of largest and smallest singular values.Thanks a lot. $\endgroup$ – Srinivas K Apr 16 '15 at 13:25
  • $\begingroup$ Both 2- and Frobenius norms are unitarily invariant; they depend only on singular values of the matrix which do not change under orthogonal transformations but do generally change if the basis change is not unitary. Positive definiteness is usually defined by having $x^*Ax$ positive for all nonzero $x$, which does not make sense if $A$ is non-Hermitian as $x^*Ax$ can be complex even if it has real eigenvalues. $\endgroup$ – Algebraic Pavel Apr 16 '15 at 13:31
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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.

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    $\begingroup$ "Minimal Polynomials are preserved for both as a result of a heap of theory." I'm not sure what this means but the argument is simple: similar matrices are similar to the same Jordan form, which determines how the minimal polynomial looks like. $\endgroup$ – Algebraic Pavel Apr 16 '15 at 9:43
  • $\begingroup$ @Algebraic Pavel: Right, but proving this requires a good deal of theoretical framework. This is what I mean, although it is true that I was needlessly cryptic. Sorry about that! $\endgroup$ – User12345 Apr 16 '15 at 14:35

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