[Math] List of matrix properties which are preserved after a change of basis

big-listlinear algebra

Lately I encountered such a problem.

Which of the properties of matrices are preserved after a change of basis ?
(orthogonal basis and square matrix are preferred in the first place)

Maybe it is a reasonable to make such a comprehensive list?
Wikipedia doesn't provide even a short list in Change of basis though it gives some answers in Matrix_similarity.
If someone knows however about such a list please give a pointer towards it. I need the list, not a bibliography.

I would like additionally to divide properties into general and specific ones, where general properties are like

symmetry, skew-symmetry, etc (they can be or not associated with a given matrix – binary decision)
and
specific properties are like
rank, determinant, trace etc.. (they can be always characterized by a single number or a set of them).

So I will start. Also important is to list what is not preserved.

What is preserved ?

General properties: (if happens)

Symmetry. Yes.

Skew-Symmetry. Yes

Orthogonality . Yes

Diagonality (non-zero entries only on diagonal) No

Positivness (all entries are positive) No.

Specific properties:

Trace. Yes.

Rank. Yes.

Determinant. Yes.

What else can be added?

….

Best Answer

Properties Preserved Under GCB:

  • Rank
  • Nullity
  • Determinant
  • Trace
  • Characteristic Polynomial
  • Eigenvalues
  • Minimal Polynomial
  • Diagonalizability
  • Jordan Canonical Form

Properties Preserved Under OCB:

  • Symmetry ($A^T=A$)
  • Antisymmetry ($A^T=-A$)
  • Orthogonality ($A^TA=I$)
  • Normality ($AA^T=A^TA$)
  • Positive (semi)definite-ness
  • Schur triangular form
  • Matrix norm (Frobenius, Euclidean)

Properties Preserved by Neither:

  • Having positive/non-negative entries
  • Image
  • Null space
  • Eigenvectors/Generalized eigenvectors
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