Eigensequences and the definition of matrix inversion

Step 1. We want to find the eigentriangle for this lower triangular Toeplitz triangle:

{A=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 2&1&0&0&0&0&0&0\\ 3&2&1&0&0&0&0&0\\   4&3&2&1&0&0&0&0\\ 5&4&3&2&1&0&0&0\\ 6&5&4&3&2&1&0&0\\ 7&6&5&4&3&2&1&0\\ 8&7&6&5&4&3&2&1\end{array}\right)

Step 2. We begin by shifting down matrix {A} one step and adding ones to the main diagonal:

{A_{down shifted}=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 2&1&1&0&0&0&0&0\\ 3&2&1&1&0&0&0&0\\ 4&3&2&1&1&0&0&0\\ 5&4&3&2&1&1&0&0\\ 6&5&4&3&2&1&1&0\\ 7&6&5&4&3&2&1&1\end{array}\right)

Step 3. We multiply matrix {A_{down shifted}} elementwise with matrix {B}

{B=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0&0\\ -1&-1&1&0&0&0&0&0\\ -1&-1&-1&1&0&0&0&0\\ -1&-1&-1&-1&1&0&0&0\\ -1&-1&-1&-1&-1&1&0&0\\ -1&-1&-1&-1&-1&-1&1&0\\ -1&-1&-1&-1&-1&-1&-1&1\end{array}\right)

Step 4. We now get matrix {C} as a result:

{C=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0&0\\ -2&-1&1&0&0&0&0&0\\ -3&-2&-1&1&0&0&0&0\\ -4&-3&-2&-1&1&0&0&0\\ -5&-4&-3&-2&-1&1&0&0\\ -6&-5&-4&-3&-2&-1&1&0\\ -7&-6&-5&-4&-3&-2&-1&1\end{array}\right)

Step 5. Calculate the matrix inverse of matrix {C} to get matrix {C^-1}:

{C^-1=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 3&1&1&0&0&0&0&0\\ 8&3&1&1&0&0&0&0\\ 21&8&3&1&1&0&0&0\\ 55&21&8&3&1&1&0&0\\ 144&55&21&8&3&1&1&0\\ 377&144&55&21&8&3&1&1\end{array}\right)

Step 6. Now by the definition of matrix inversion discussed here: The inverse of a triangular matrix using forward substitution, we have: {C*C^-1=I} where {I} is the identity matrix. Written as matrices we then can write:

\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0&0\\ -2&-1&1&0&0&0&0&0\\ -3&-2&-1&1&0&0&0&0\\ -4&-3&-2&-1&1&0&0&0\\ -5&-4&-3&-2&-1&1&0&0\\ -6&-5&-4&-3&-2&-1&1&0\\ -7&-6&-5&-4&-3&-2&-1&1\end{array}\right){*}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 3&1&1&0&0&0&0&0\\ 8&3&1&1&0&0&0&0\\ 21&8&3&1&1&0&0&0\\ 55&21&8&3&1&1&0&0\\ 144&55&21&8&3&1&1&0\\ 377&144&55&21&8&3&1&1\end{array}\right){=}\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\end{array}\right)

Step 7. Write matrix {C} again and look at the identity for the first column of {C^-1}

\left(\begin{array}{cccccccc}  1&0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0&0\\ -2&-1&1&0&0&0&0&0\\ -3&-2&-1&1&0&0&0&0\\ -4&-3&-2&-1&1&0&0&0\\ -5&-4&-3&-2&-1&1&0&0\\ -6&-5&-4&-3&-2&-1&1&0\\ -7&-6&-5&-4&-3&-2&-1&1\end{array}\right){*}\left(\begin{array}{cccccccc}1\\ 1\\ 3\\ 8\\ 21\\ 55\\ 144\\ 377\end{array}\right){=}\left(\begin{array}{cccccccc}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right)

Step 8. Multiplying the expression on the left of the equal sign we get:

\left(\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0&0\\ -2&-1&3&0&0&0&0&0\\ -3&-2&-3&8&0&0&0&0\\ -4&-3&-6&-8&21&0&0&0\\ -5&-4&-9&-16&-21&55&0&0\\ -6&-5&-12&-24&-42&-55&144&0\\ -7&-6&-15&-32&-63&-110&-144&377\\\end{array}\right){=}\left(\begin{array}{cccccccc}1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right)

Step 9. Sum the terms on the left side of the equal sign and notice that they equal the right hand side of the equal sign.

Step 10. Remove the elements in the main diagonal to get matrix {D}:

{D=}\left(\begin{array}{cccccccc}0&0&0&0&0&0&0&0\\ -1&0&0&0&0&0&0&0\\ -2&-1&0&0&0&0&0&0\\ -3&-2&-3&0&0&0&0&0\\ -4&-3&-6&-8&0&0&0&0\\ -5&-4&-9&-16&-21&0&0&0\\ -6&-5&-12&-24&-42&-55&0&0\\ -7&-6&-15&-32&-63&-110&-144&0\end{array}\right)

Step 11. Change the signs of the elements in matrix {D} to get matrix {E}:

{E=}\left(\begin{array}{ccccccc}1&0&0&0&0&0&0\\ 2&1&0&0&0&0&0\\ 3&2&3&0&0&0&0\\ 4&3&6&8&0&0&0\\ 5&4&9&16&21&0&0\\ 6&5&12&24&42&55&0\\ 7&6&15&32&63&110&144\end{array}\right)

Matrix E is called an eigentriangle because the last elements in each row equals the row sums of the previous row. This property of the eigentriangle is because of the definition of matrix inversion which was illustrated above. 1, 1, 3, 8, 21, 55, 144, 377 is called an eigensequence.

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