# Show that diagonally dominated matrices are always invertible. Now, what is a diagonally dominated matrix? It is a square matrix that has each of it's diagonal values larger in magnitude than all the other values in the rows combined.
# Show that diagonally dominated matrices are always invertible. Now, what is a diagonally dominated matrix? It is a square matrix that has each of it's diagonal values larger in magnitude than all the other values in the rows combined.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Kiti laijlla| < (Ei+j lai,j)|x;| using (2).
Hints:
• Remember that, a matrix A is invertible if the only solution to Ax = 0 is the 0 vector, i.e.
x = 0.
• Suppose, for a DDM (diagonally dominated matrix), there exists a non-zero solution to
X1
X2
Ax = 0. Since x =
Let's say, xi has the greatest magnitude here which means |x;| > |xj| for all j # i ...(2).
• Since, Ax = 0, we must have,
ai,1X1+ ai,2x2 + · · · + ai.nxn = 0 for the i'th row.
In other words,
ai,1X1 + ai,2x2 + · · ·+ ai,i–1Xi-1+ai,i+1Xi+1+ • · · + ai,nXn = –aj,iXi
or,
|ai,121 + ai,2x2 + + a;,i-1Xi-1+a¿,i+1Xj+1+ . . + ai,n&n| = |ai,¿X¡|
But we will prove that, this is not possible when x is non-zero.
So, we have proved that, |Eitj aijaj| < |ai,iXi|
Explanation:
|Eitj ai,ja;| < Eiti lai,j*;| because |æ + y| < |x| + \y| for all real numbers x, y, z.
¿ lai.jª j| = Eitj lai.j||xj| because |æy| = |c||y| for all real a, y.
(Eitj laij)|#;| < |a;,a||x:| using (1)
Transcribed Image Text:# Show that diagonally dominated matrices are always invertible.
Now, what is a diagonally dominated matrix? It is a square matrix that has each of it's diagonal
values larger in magnitude than all the other values in the rows combined.
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