Complete Invariant

What is a complete invariant?

A complete invariant is a property that remains unchanged under a given transformation.

Understanding complete invariants is extremely useful, as it allows us to determine whether two objects are equivalent, or to identify the nature of a mathematical or geometric transformation.

Examples. In linear algebra, both the determinant and the rank of a matrix are complete invariants. Similarly, the distance between two points in the Cartesian plane remains unchanged under rotations or translations, among other examples.

A practical example

Example 1

The determinant is a complete invariant under the transposition of a square matrix, as it remains unchanged when the matrix is transposed.

Consider the following matrix:

$$ M = \begin{pmatrix} 1 & 4 \\ 5 & 2 \end{pmatrix} $$

The determinant of this matrix is:

$$ \det \begin{pmatrix} 1 & 4 \\ 5 & 2 \end{pmatrix} = 1 \cdot 2 - 4 \cdot 5 = 3 - 20 = -17 $$

The transpose of matrix M is:

$$ M^T = \begin{pmatrix} 1 & 5 \\ 4 & 2 \end{pmatrix} $$

The determinant of the transposed matrix M^T remains -17:

$$ \det \begin{pmatrix} 1 & 5 \\ 4 & 2 \end{pmatrix} = 1 \cdot 2 - 5 \cdot 4 = 3 - 20 = -17 $$

Example 2

The rank of a matrix is also a complete invariant under transposition, as it remains unaffected.

For example, consider the matrix M:

$$ M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$

The rank of this matrix is:

$$ r(M)=2 $$

Now, let’s compute the transpose of M:

$$ M^T = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix} $$

The rank of the transposed matrix M^T remains 2:

$$ r(M)=2 $$

And so on.