os A=mo :matrix thats is symmetric & pos. Tefinite of then is A singular? non
please explain in full
details.
A square matrix A is skew-symmetric if A = -A (a) If A is an n xn skew-symmetric matrix, with n odd, prove that A is singular, i.e. non-invertible (b) Find a skew-symmetric matrix that is invertible.
Consider the singular value decomposition (svd) of a symmetric matrix, A- UAU Show that for any integer, n, An-UNU. Argue that for a psd matrix A, there must exist a square root matrix, A-such that 1/2 1/2 A 1/2
Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes.
Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes.
3. Answer the following questions regarding positive definite matrix. A symmetric real matrix M is said to be positive definite if the scalar 27 Mz is positive for every non-zero column vector z (a) Consider the matrix [9 6] A = 6 a so that the matrix A is positive definite? What should a satisfy (b) Suppose we know matrix B is positive definite. Show that B1 is also positive definite. Hint use the definition and the fact that every...
Determine which of the following matrices are (1) symmetric, (ii) singular, (iii) strictly diagonally dominant, (iv) positive definite. 2 0 0 1 3 0 0 0 4 symmetric [Choose] Singular [Choose] strictly diagonally dominant Yes Yes positive definite matrix [Choose]
Let A E Mn(R) be a non-singular matrix. Show that if λ 1/λ is an eigenvalue of A-1 0 is an eigenvalue of A, then
2. Write the product of a sequence of elementary matrices which equals the given non-singular matrix: [ 11 2 3 3. Given the matrix A = 01 - write the matrix of minors of A, the matrix of cofactors of A, the adjoint 12 2 2 matrix of A, and use the adjoint of A, to write the inverse of A. 4. Determine whether the set of vectors is linearly dependent or linearly independent. Justify your answer. 13
1
2
Suppose that the non-singular n × n matrix A can be diagonalized, ie A = PDP-1 where D is a diagonal matrix. Show that A-1 and AT can be diagonalized. 1.e. Suppose we have 2nu x 2n block matrices Y=I-I B O AB where all sub-matrices are n × n and O denotes the zero matrix. Find a block matrix X such that XY the determinant of X? Z and demonstrate it works. What is
true or false and explain why
(a) If the eigenvalues of a real symmetric matrix Anxn are all positive, then 7" A7 > 0 for any i in R" (b) If a real square matrix is orthogonally diagonalizable, it must be symmetric. (c) If A is a real mx n matrix, then both APA and AA' are semi-positive definite. (d) SVD and orthogonal diagonalization coincide when the real matrix concerned is symmetric pos- itive definite. (e) If vectors and q...
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix...