3. Consider the matrix $A = \begin{pmatrix} 3 & 3 \\ -1 & 1 \end{pmatrix}$ (a) Compute $A^T A$ and its eigenvalues and unit eigenvectors $q_1$ and $q_2$ and compute $\sigma_1$. (b) Compute $AA^T$ and its eigenvalues and unit eigenvectors $p_1$ and $p_2$. (c) Put the numbers into the format $A = P \Sigma Q^T$ (d) Show that $A = \sigma_1 p_1 q_1^T + \sigma_2 p_2 q_2^T$. What is the significance of this decomposition? (i.e. if A has rank r then how do you interpret the matrix A?). (e) From the q's and p's write down the orthonormal bases for the four fundamental subspaces of the matrix A. (f) Describe all the matrices that have those same four fundamental subspaces. Are the multiplies of A? (g) Compute the pseudo-inverse $A^+$. Does the pseudo-inverse equal to another matrix? (h) Rename the four fundamental subspaces (for A) as four subspaces of $A^+$. (i) Compute the projections $AA^+$ and $A^+A$. What do these projections project onto?
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$A^T A = \begin{pmatrix} 3 & -1 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} 3 & 3 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 10 & 8 \\ 8 & 10 \end{pmatrix}$ $AA^T = \begin{pmatrix} 3 & 3 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 3 & -1 \\ 3 & 1 \end{pmatrix} = Show more…
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Consider the 2 !! 4 matrix A = [0 0 3 0; 1 0 0 1] and let r = rank(A). We will work our way toward the Singular Value Decomposition (SVD) of A and to the pseudoinverse of A. a.) Briefly explain why we know, without having to do any computations, that A^T A has at least two eigenvalues equal to 0 b.) Calculate A^T A. [Hint: Only five entries are nonzero] c.) Show that det(A^T A – λ I) = λ^4 – 11λ^3 + 18λ^2. d.) Compute the eigenvalues of A^T A. Label and order them as λ1 ≥ λ2 ≥ λ3 ≥ λ4. e.) Compute the singular values of A. Label and order them as σ1 ≥ σ2 ≥ σ3 ≥ σ4. f.) What is the matrix Σ1 for the reduced SVD of A? What is the matrix Σ for the SVD of A? g.) For all of the nonzero eigenvalues of A^T A, find corresponding unit eigenvectors. h.) Compute rref(A^T A) and find the "special solutions". i.) Apply the Gram-Schmidt Process to the vectors you found in part (h). j.) What is the matrix V1 for the reduced SVD of A? What is the matrix V for the SVD of A? k.) For i = 1, . . . , r, compute the vectors ui. l.) For i = r + 1, . . . , m, compute the vectors ui. If there are no such vectors, briefly explain why. m.) What is the matrix U1 for the reduced SVD of A? What is the matrix U for the SVD of A? n.) What is the matrix Σ+ for computing the pseudoinverse of A? o.) Which matrix multiplication would you perform to compute the pseudoinverse of A? (Do not perform the multiplication)
Supreeta N.
2. Let Ε = {E, E, E, E} be the standard ordered basis for M₂ₓ₂(ℑ), where M₂ₓ₂(ℑ) is the vector space of all 2 2 real matrices under the usual matrix addition and scalar multiplication. Let T: M₂ₓ₂(ℑ) → M₂ₓ₂(ℑ) the linear operator defined by T(B) = Bᅁ, the transpose of B, for each B ∈ M₂ₓ₂(ℑ) (a) (4 pts) Use definition to compute matrix A = [T]ᄉ. Determine all values λ such that det(A − αI) = 0, where I is the identity 4 4 matrix. Show your work. (b) (8 pts) For each eigenvalue λ obtained in (a), find its corresponding eigenspace Eᅁ = N(A − αI) = {x ∈ ℑ⁴ : (A − αI)x = 0} and determine a basis for Eᅁ. Show your work. (c) (3 pts) Use (b) to show that there exists a basis β for ℑ⁴ such that Q⁻AQ is diagonal matrix for some invertible matrix Q as in Theorem 2.23 (Section 2.5 on Page 113) and the paragraph after Example 6 in Section 5.1 on Pages 251-252.
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A = [-1 1 2; 1 2 1; 2 1 -1] 1) Find tr A, tr2 A, det A and det(A-λ I) 2) Find eigenvalues λ1 ≤ λ2 ≤ λ3 3) Find eigenvectors Avj = λj Vj, j=1,2,3. 4) For vectors f = (0, -4, 2)T solve the initial value problem ∂tu=Au, u(0)=f 5) Find orthogonal matrix U such that A=UDU-1, D=[λ1 0 0; 0 λ2 0; 0 0 λ3] 6) On subspace S = Span {V1, V2} for any g∈S find lim etAg as t→+∞ 7) Find orthogonal projection of vector f=(0,-4,2)T onto subspace S
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