Let
be a
matrix with 3 different eigen values. One of them has multiplicity
3, then the other two have 1.
Let
has multiplicity 3 and
has multiplicity 1.
Now for
we will definitely get 1 eigen vector each that is total 2.
and now we have,
has rank 2. Then the number of eigen vector for
is 2.
Therefore, the total number of eigen vector is 4 (<5)
Hence, The matrix
is defective.
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let 2 be an...
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let a be an eigenvalue of multiplicity 3. If A-21 has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let i be an eigenvalue of multiplicity 3. If A-11 has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
(b) The matrix B= 1 2 2 3 3 1 3 2 4 has eigenvalues 7,2, -1. i. Find a column and a row eigenvector of B corresponding to the Perron eigenvalue. ii. Find a rank one nonnegative matrix C such that the matrix B+C will have eigenvalues 13, 2, -1. iii. Let a and B be real numbers. Calculate the eigenvalues of D(a, b) = aB+ BC. iv. Find limno(+B)"
(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that tr(A) = 1 and det(A) = 63. Find the eigenvalues of A with their algebraic multiplicities. The smaller eigenvalue has multiplicity and the larger eigenvalue has multiplicity
9. Let Q1 S Q2 be the eigenvalues of the matrix A and Qs 1 if Qi is a defective eigenvalue and Q. - 0 otherwise, where 86 100 01-81-94 T < 1 Let Q = 1013 + IQ1 + 2(Qal + 3(Qal). Then T = 5 sin2(100Q) satisfies:-(A) 0 (B) 1 T < 2.
9. Let Q1 S Q2 be the eigenvalues of the matrix A and Qs 1 if Qi is a defective eigenvalue and Q. - 0...
LINEAR ALGEBRA
What are the eigenvalues of the matrix 2-31 1 -2 1 What is the characteristic polynomial of this matrix? (That is, the polynomial you use to find the eigenvalues). 1-32 p() = -13 +212 -1. op() = X(4-1)(-2) op() = 12 + 1 OPW) = 12 -2X + 1. Let M = 2-31 1 -2 1 1-32 (this is the same as the previous problem). Find the eigenvalues of M (they are not listed according to multiplicity). Let...
hi, can you write the notation
and everything?
Assignment 12 Let A= -3 -5 -2 6 8 2 0 0 -2 [3pt] (a) Find the eigenvalues of A. (1pt] (b) Determine the algebraic multiplicity of each eigenvalue of A. [3pt] (c) Find a basis for each eigenspace for A. (1pt] (d) Determine the geometric multiplicity of each eigenvalue of A. [2pt] (e) Give a matrix P and a diagonal matrix D such that P-1AP = D.
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
For the 3×2 matrix A:
a) Determine the eigenvalues of ATA, and confirm that
your eigenvalues are consistent with the trace and determinant of
ATA.
b) Find an eigenvector for each eigenvalue of
ATA.
c) Find an invertible matrix P and a diagonal matrix D such that
P-1(ATA)P = D.
d) Find the singular value decomposition of the matrix A; that
is, find matrices U, Σ, and V such that A = UΣVT.
e) What is the best rank 1...
3 1. Let A = 0 (a) Compute the eigenvalues of A and specify their algebraic multiplicities. (b) For every eigenvalue 1, determine the eigenspace Ex and specify its dimension. (c) Is A a defective matrix? Why or why not? (d) Is A a singular matrix? Why or why not? (e) Determine the eigenvalues of (74) + 5.