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Restricted Diagonalisation

Suppose that row operations and column operations are restricted in a way that

Row Operations:

(R1) Add an integral multiple of a row to another.

(R2) Exchange two rows.

(R3) Multiply a row by -1 .

Column Operations:

(C1) Add an integral multiple of a column to another.

(C2) Exchange two columns.

(C3) Multiply a column by -1 .

With such rules, is there a mean for us to diagonalise

say \( \begin{bmatrix} a & b &c \\ d&e & f\\ g& h & i \end{bmatrix} \), that is of integral elements, to \( \begin{bmatrix} a' & b' &0 \\ d'&e' & 0\\ 0& 0 & i' \end{bmatrix} \), that is also of integral elements only, without resorting to Gaussian Elimination?

Note by L Km
5 months, 1 week ago

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The rules of the restricted row operation can be thought as be the subset of the rules of normal row operations,

in rules of normal row operations, say in \( (R1), \) we can multiply an multiple of a real number and one of the row to another row.

Besides, the restricted operations are rules on the manipulation on diophantine equation, based on the fact that \( \begin{cases} & a_1x+a_2y+a_3z= n_1 \\ & b_1x+b_2y+b_3z= n_2 \end{cases}\Leftrightarrow \begin{cases} & k_1(a_1x+a_2y+a_3z)+ k_2 (b_1x+b_2y+b_3z)= k_1 n_1 +k_2 n_2 \\ & -1( b_1x+b_2y+b_3z )= -n_2 \end{cases} \) L Km · 5 months, 1 week ago

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@L Km Great! So you identified that "We changed the condition from real multiples to integer multiples". We know that for real multiples, this can always be done. We want to know what happens for integer multiples.

So, that's a great place to start. Think of a system which requires you to use real, but not integer, multiples. Does that provide a counter example?
If yes, why? How can we explain that this is impossible?
If no, why not? How did we circumvent it? Calvin Lin Staff · 5 months, 1 week ago

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What's the difference between your set of rules, and the rules of Gaussian elimination?

That will provide some insight into which steps of Gaussian are invalid, and then we can hope to fix them. If it turns out that we cannot fix them, that might provide an idea of finding a counter example. If we can fix all the steps, then the answer would be yes :) Calvin Lin Staff · 5 months, 1 week ago

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