Question

A = [-1 1 2; 1 2 1; 2 1 -1] 1) Find tr A, tr2 A, det A and det(A-? I) 2) Find eigenvalues ?1 ? ?2 ? ?3 3) Find eigenvectors Avj = ?j 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=[?1 0 0; 0 ?2 0; 0 0 ?3] 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

          A = [-1 1 2; 1 2 1; 2 1 -1]

1) Find tr A, tr2 A, det A and det(A-? I)
2) Find eigenvalues ?1 ? ?2 ? ?3
3) Find eigenvectors Avj = ?j 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=[?1 0 0; 0 ?2 0; 0 0 ?3]
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

A = [-1 1 2; 1 2 1; 2 1 -1]

1) Find tr A, tr2 A, det A and det(A-? I)
2) Find eigenvalues ?1 ? ?2 ? ?3
3) Find eigenvectors Avj = ?j 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=[?1 0 0; 0 ?2 0; 0 0 ?3]
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

Added by Roy B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

05/16/2024

Step 1

The matrix A is given by: A = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix} The trace of A is the sum of the diagonal elements: tr(A) = 1 + 2 = \boxed{3} The square of A is: A^2 = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 2 & 2 Show more…

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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