Prove 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. To be mathematically precise, for every diagonal value a;,i,lai,i| > Eitj lai,j| ...(1)|ai,il > Jai,i[+ |ai,2| + ...For example,[3+ |ai,i-1|+ |ai,i+1|+....+ lai,n|115 2113126.-3113114Hints:• 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|x1X2Ax= 0. Since x =Let's say, x; has the greatest magnitude here which means |x;|> |xj| for all j #i . .(2).• Since, Ax = 0, we must have,Ai,1X1+ ai,2X2+ • · ·In other words,+ ai,nxn = 0 for the i’th row.ai,1x1+ a¿.2x2+ • · ·+ ai,i–1xi–1+ai,i+1Xi+1 +·…+ ai,n Xn = -ai,ix;or,|ai,1x1 + ai,2X2 + · · ·+ ai,i-1Xi-1+ai,i+1Xi+1+•• •+ ai,nºn| = |ai,iXi|TEitj di,j¤j| = |a;,i||¤||But we will prove that, this is not possible when x is non-zero.itjSo, we have proved that, |Eit; ai,ja;| < |ai,iX;|Explanation:IEit; ai,jxj| < Eit; lai,j®;| because |æ + y| < |x| + ly| for all real numbers r, y, z.Eitj lai,jaj| = Eit¡lai,j||x;| because |xy| = |x||y| for all real r, y.Eits laij||a;| < (it; lai,j)|#:| using (2).(Eitj lai,j))|æ:| < |a;,i||¤:| using (1)

Prove that diagonally dominated matrices are always invertible. A square matrix is said to be…

Prove 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.

Prove 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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