Are the singular values of $A^2$ necessarily the same as the squares of the singular values of A? (either find a counterexample by hand or with julia, or prove that it is always the case, or demonstrate with enough examples to be convicing with julia)
Are the singular values of $A^TA$ necessarily the same as the squares of the singular values of A? (either find a counterexample by hand or with julia, or prove that it is always the case, or demonstrate with enough examples to be convicing with julia)
A = [ 1 4 2;2 8 4; -1 -4 -2]
3×3 Array{Int64,2}: 1 4 2 2 8 4 -1 -4 -2
using LinearAlgebra
U,s,V =svd(A, full=true)
display(U)
display(s)
display(V)
3×3 Array{Float64,2}: -0.408248 0.912871 7.81735e-17 -0.816497 -0.365148 0.447214 0.408248 0.182574 0.894427
3-element Array{Float64,1}: 11.22497216032183 4.845410522502476e-16 0.0
3×3 Adjoint{Float64,Array{Float64,2}}: -0.218218 -0.9759 0.0 -0.872872 0.19518 -0.447214 -0.436436 0.09759 0.894427
4a. What is the rank of this matrix?
4b. For which right hand sides is Ax=b solvable? (Find a condition on b₁,b₂,b₃)?
A = [1 4; 2 9;-1 -4]
3×2 Array{Int64,2}: 1 4 2 9 -1 -4
using LinearAlgebra
U,s,V =svd(A, full=true)
display(U)
display(s)
display(V)
3×3 Array{Float64,2}: -0.377924 0.59764 0.707107 -0.84519 -0.534466 -1.38778e-15 0.377924 -0.59764 0.707107
2-element Array{Float64,1}: 10.907941643728067 0.12964990174715935
2×2 Adjoint{Float64,Array{Float64,2}}: -0.224261 0.974529 -0.974529 -0.224261
5a. What is the rank of this matrix?
5b. For which right hand sides is Ax=b solvable? (Find a condition on b₁,b₂,b₃)?
(6) Explain why the set of singular matrices is not a subspace.
(7) If the 9x12 system Ax=b is solvable for every b then the column space of A is .......?
(8) GS p143 3.2 15 done with the svd on a computer:
Construct a matrix for which N(A) = all combinations of (2,2,1,0) and (3,1,0,1)
Step 1: Find an orthogonal matrix whose first two columns are linear combinations of the given vectors:
Notice that we input a 4x2 matrix but Julia's QR returns a complete square orthgonal matrix whose first two columns are the Q we saw in class.
using LinearAlgebra
N = [2 3
2 1
1 0
0 1];
Q, = qr(N) # we don't need R just the "Q"
W = Q[:,[3,4]] # take the last two columns of Q ( " [3,4] " means take column 3 and 4, note that the commas are needed)
4×2 Array{Float64,2}: 0.11684 -0.397212 -0.527969 0.348208 0.822258 0.0980073 0.17745 0.843427
Step 2: W' immediately gives a right answer. Let's check this.
W'N
2×2 Array{Float64,2}: -1.11022e-16 -1.66533e-16 -8.32667e-17 3.33067e-16
Understanding that the last two columns of Q are the completion of the left part to an orthogonal matrix explain why this worked.
(9) (Julia submit a screenshot problem) GS p143 3.2 16:
With Julia construct A so that the nullspace of A = all multiples of (4,3,2,1). Its rank is ..... ?
using LinearAlgebra
N = [4
3
2
1]
# Please finish the computation following problem 8 as a template
4-element Array{Int64,1}: 4 3 2 1
# please provide a screenshot of your check
(10) Use the svd to explain why no 3x3 matrix have a nullspace that equals its column space.