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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 30E

See similar textbooks

Related questions

Q: MAM™ is similar to A if and only if M is O a. Symmetric matrix O b. Invertible matrix O c Orthogonal…

Q: An mxn upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown…

Q: Show that the matrix is not diagonalizable.

A: Solve for det(A - λI) = 0

Q: Determine whether the given matrix is diagonalizable 3 1 A = 0 1 3

Q: Prove that a matrix that is both unitary and upper triangular must be a diagonal matrix.

A: Given that the matrix A is both unitary and upper triangular.

Q: Determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is…

A: Given matrix is

Q: what is a Hermitian matrix? does every Hermitian matrix has to be square matrices?

Q: Explain why the matrix A + C does not exist.

Q: Find a sequence of elementary matrices whose product is the given nonsingular matrix.

Q: If A is an invertible matrix, explain why A and AT must have the same null space. Is this true if A…

A: If A is invertible matrix then it has trivial null space and a matrix is invertible iff AT is…

Q: Prove that any square matrix A can be written as the sum of a symmetric matrix and a skewsymmetric…

A: To Prove Any Square matrix A can be written as the sum of a symmetric matrix and a skewsymmetric…

Q: Prove that the main diagonal of a skew-symmetric matrix must consist entirely of zeros.

A: Given-Skew symmetric matrix. To prove- That the main diagonal of a skew-symmetric matrix must…

Q: Is the matrix diagonalizable? If yes, find diagonal matrix D and regular matrix P such that AP = PD.…

A: Yes, Given Matrix is diagnolizable.

Q: Disprove that the rank of a matrix is equal to the number of nonzero rows.

Q: Let A be a 3 x 5 matrix in which not all entries are zero. (a) What size must the matrix B be for AB…

Q: Is it possible for a matrix to be both upper triangular and lower triangular? Oa. Yes O b. No

A: Solution

Q: Prove that every symmetric upper (or lower) triangular matrix is diagonal.

A: Solution: Consider the given statement " Every symmetric upper (lower) triangular matrix is…

Q: Prove that if an upper triangular matrix is orthogonal, then it must be a diagonal matrix

A: Solution: If the multiplication of matrix with its transpose is equal to the identity, then a matrix…

Q: Determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is…

Q: Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type

A: Let A be a non-singular matrix of order n. Then, A can be reduced to identity matrix In by a finite…

Q: Show that any nX n matrix can be written as the sum of a symmetric matrix and a skew-symmetric…

A: To show that any n x n matrix can be written as the sum of a symmetric matrix and a skew-symmetric…

Q: Let A and B be two square Matrices of the same Size such that A is not invertible (B may or not be…

Q: Determine whether the matrix 1 0 0 4 0 0 0 0 0 1. A = 4 1 4 is diagonalizable. If not, explain why…

A: See the detailed solution below.

Q: Show that if A is a symmetric nonsingular matrix then A−1 is also symmetric.

A: A is a symmetric non singular matrix, which means inverse exists and

Q: Prove that an upper triangular n ×n matrix is invertible if and only if all its diagonal entries are…

Q: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

A: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

Q: TRUE or FALSE: Every invertible n xn matrix is similar to the identity matrix In.

A: To state true or false given statement.

Q: Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal…

A: Permuting the diagonal entries of a diagonal matrix D amounts to permuting the rows of that matrix…

Q: Prove that an upper or lower triangular n x n matrix is invertible if and only if all its diagonal…

A: Consider A be a n x n upper or lower triangular matrix

Q: Let A and B be n x n matrices such that AB is invertible. Prove that A and B are invertible. Give an…

Q: Consider the following matrix: A-() 1. Determine whether A is diagonalizable over C. If so,…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: If A is a symmetric matrix, does it follow that the transpose of matrix A is also symmetric?…

Q: Suppose A is a 10x10 invertible matrix. Then A is row equivalent to the 10x10 identity matrix. A)…

Q: Prove that the product of two upper triangular matrices is always another upper triangular matrix.

A: We have to prove that the product of two upper triangular matrices is always another upper…

Q: If A is invertible, then the columns of A 1 are linearly independent. Explain why. A. If A is…

A: Option (D) is the correct option.

Q: Determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is…

Q: Show that if the matrix X is diagonalizable, then X" is also diagonalizable.

A: Given that the matrix X is diagonalizable. We have to show that XT is also diagonalizable. Since,…

Q: # Show that diagonally dominated matrices are always invertible. Now, what is a diagonally dominated…

Q: Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible.

Q: Prove that if a symmetric matrix is invertible, then its inverse is symmetric also

A: Let A be a invertible symmetric matrix. To prove that, Inverse of a symmetric matrix is symmetric.…

Q: Give an example of a 4 by 4 matrix A, which is not symmetric, however A is symmetric.

A: A square matrix is said to be symmetric if it is symmetric about the main diagonal.

Q: Prove that if both products AB and BA are defined, then AB and BA are square matrices. Assume that A…

A: AB and BA are both defined.

Q: Prove that diagonally dominated matrices are always invertible.

A: Suppose A be a n×n diagonally dominated matrix. Therefore: aii>∑i≠jaij. Suppose A is not…

Q: If AC is an invertible matrix, then show that A is also an invertible matrix

A: Given- AC is an invertible matrix, i.e.- ACAC-1=I⇒ACA-1C-1=I⇒AA-1CC-1=I

Q: a13 a22 > a21 + a23 a33 > a31 + a32 Note: a11, a22, and a33 should be greater than (>) and not…

A: Introduction: A square matrix is said to be diagonally dominant if the magnitude of the diagonal…

Q: Prove that every 3 x 3 matrix is a linear combination of invertible matrices.

A: A square matrix is invertible if and only if the determinant of the matrix is not equal to zero.…

Q: A matrix which is not a square matrix is called a .6 Matrix

A: 6) A matrix which is not a square matrix is called a

Q: Prove that the sum and the product of two lower- (upper-) triangular matrices is lower- (upper-)…

A: Topic:- matrix

Question

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|
1
1
5 2
1
1
3
1
2
6.
-3
1
1
3
1
1
4
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, 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.
itj
So, 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)

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

The Segment-Addition Postulate can be generalized as follows: The length of a line segment equals the sum of the length of its parts. State a general conclusion regarding AE based on the following figure.
Let all of the lines named be coplanar. Make a drawing to reach a conclusion. a If rs and st, then _. b If ab and bc, then _.
Prove that the altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.
a Does the similarity relationship have a reflexive property for triangles and polygons in general? b Is there a symmetric property for the similarity of triangles and polygons? c Is there a transitive property for the similarity of triangles and polygons?
To prove a theorem of the form "If P, then Q" by the indirect method, the first line of the proof should read: Suppose that ___________ is true.
For what values of c is c(1,2,3)=1?
Complete an analytic proof of the following theorem: In a triangle that has sides of lengths a,b, and c, if c2=a2+b2, then the triangle is a right triangle.
Show that u and v in the figure at the right are equivalent.