List of matrix properties which are preserved after a change of basis

Question

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.

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What else can be added?

....

Answer 1

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