For , the trace is

It is linear and cyclic: , and for any conformable matrices

When R is a field— or embeds in a field F—then over an algebraic closure of ) equals the sum of the eigenvalues of A counted with algebraic multiplicity; see [11].

The anti-trace is the sum of the anti-diagonal entries,

Let be the exchange (reversal) permutation matrix with ones on the anti-diagonal and zeros elsewhere. Then and and

For , so , consistent with [10].

Given a univariate sequence , its ordinary generating function (OGF) is 0 } f _ { n } x ^ { n } \end{document} ]]> . For a three-term linear recurrence with fixed the OGF satisfies

We will also use trivariate OGFs of the form together with the multinomial coeficients

All generating functions are formal power series (no convergence issues). These tools let us encode and solve the coeficient recurrences appearing later.

Using (4) we compute the first ten values of

The pattern suggests

with depending only on

• Term

Using equation (4), the coeficients of on both sides implies

where . Define a generating function as follows

Accordingly, we obtain

The above equation is true since for the function satisfies

Therefore, by equation (6) it implies

Then, it follows that

As a conclusion, the function can be written as

• Term

Without loss of generality, the same technique is applied in order to find

. It yields a similar expression as ,k.

• Term

Interestingly, using the same method for finding difers slightly from the other cases. By comparing the coeficient of on both sides and using equation (4), this implies for

Define the following generating function

Consequently, it results

The above equation is true since for the function satisfies

Therefore, by equation (9) it implies

Then, it follows that

As a conclusion, the function can be written as

For example, if , then the coeficient of is 9 in and this is true based on Table 1.

To sum up, by substituting equations (7), (8), and (10) into equation (5), hence the explicit formula for anti-trace is obtained and stated in the following theorem.

Theorem 3.1. For matrix A and positive integer n,

□

Example 3.2.Let

Then

and the anti–principal sums are

Since A is upper triangular with (1, 3) entry 2, one checks directly that

hence

We now compare with the theorem.

• For , the table/recurrence gives

which matches

• For

which matches

We investigate the anti–trace of powers of block–anti–diagonal 3m × 3m matrices with blocks.

Theorem 3.3 (Even number of blocks). Let m be even and

Then for all

Proof. A direct block multiplication shows

Hence is block–diagonal for every n. When m is even, the global anti–diagonal of the m × m block grid passes between diagonal blocks,

Then every entry on the (full) anti–diagonal of is zero; thus

For odd powers, we have

As we can see, has nonzero blocks exactly on the block anti–diagonal:

Along the full matrix anti–diagonal, one picks out the anti–diagonals of these blocks yield □

Example 3.4 (Even number of blocks). Take and

Then

Because m is even, the global anti–diagonal of passes through of–diagonal blocks only; hence . In particular,

For odd powers,

Since the anti–trace of a diagonal matrix diag equals , we get

exactly as predicted by Theorem 3.3 (ii) with

Theorem 3.5 (Odd number of blocks). Let m be odd and . With A as above, for all

Proof. As before, is obtained via direct block multiplication,

When m is odd, the global anti–diagonal meets only the central diagonal block, so reduces to the anti–trace of

For odd powers, we have

So, has nonzero blocks on the block anti–diagonal with

Taking the sum of the anti–traces of these blocks along the full anti–diagonal gives

Example 3.6 (Odd number of blocks). Let with central index and

Then so the global anti–diagonal meets only the middle block. Therefore

confirming Theorem 3.5 (i) for

For odd powers,

and hence, using 2

which matches Theorem 3.5 (ii) with

For a single numerical instance , computing by exponentiation by squaring and then taking is typically faster than evaluating the closed form. matrix power by squaring uses Θ(log n) matrix multiplications; each multiplication costs a fixed number of scalar operations (e.g., 27 multiplies and 18 additions in the naive scheme), so the overall cost is Θ(log n).

By contrast, the closed form

has terms. Indeed, the number of triples with is

so a direct evaluation of the multinomial sums scales quadratically in n (after a onetime extraction of from . This explains the observed timings: for a single large n with floating-point entries, the direct method is usually faster.

However, the closed form remains computationally useful in exact and modular arithmetic. Over general commutative rings , or polynomial rings), the recurrence/closed form permits exact computations and congruence arguments; forming numerically would either be impossible or unreliable. The structural block anti-diagonal identities in Subsection 3.2 also reduce large problems to blocks, which a direct power need not reveal.

In summary, for a single floating-point instance the direct method is often the fastest , while the closed form (and its induced recurrence) is preferable for symbolic/modular settings, when many values are required, or when exploiting block structure.