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Problem 3 Basic linear algebra related to eigenvalues Let A = egin{pmatrix} 1 & 0 \ 0 & -1 end{pmatrix}, and B = egin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}. State if each of the following is TRUE or FALSE for the above matrices A and B. In each case, you must provide reasons/hand calculations to support your answer. (a) The matrices A and B have same eigenvalues. (b) Eigenvalues of AB are the product of the respective eigenvalues of A and the eigenvalues of B, that is, ?_i(AB) = ?_i(A)?_i(B), for i = 1, 2. (c) ?_i(A + B) = ?_i(A) + ?_i(B), for i = 1, 2. (d) Consider the elementwise product A ? B (this is same as the MATLAB "dot star" operation A .* B). Then ?_i(A ? B) = 0, for i = 1, 2. (e) Consider the Kronecker product A ? B = egin{pmatrix} a_{11}B & a_{12}B \ a_{21}B & a_{22}B end{pmatrix}. Then | ?_i(A ? B) | = 1, for i = 1, ..., 4.

          Problem 3

Basic linear algebra related to eigenvalues

Let

A = egin{pmatrix} 1 & 0 \ 0 & -1 end{pmatrix}, and B = egin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}.

State if each of the following is TRUE or FALSE for the above matrices A and B. In each case, you must provide reasons/hand calculations to support your answer.

(a) The matrices A and B have same eigenvalues.

(b) Eigenvalues of AB are the product of the respective eigenvalues of A and the eigenvalues of B, that is, ?_i(AB) = ?_i(A)?_i(B), for i = 1, 2.

(c) ?_i(A + B) = ?_i(A) + ?_i(B), for i = 1, 2.

(d) Consider the elementwise product A ? B (this is same as the MATLAB "dot star" operation A .* B). Then ?_i(A ? B) = 0, for i = 1, 2.

(e) Consider the Kronecker product A ? B = egin{pmatrix} a_{11}B & a_{12}B \ a_{21}B & a_{22}B end{pmatrix}. Then | ?_i(A ? B) | = 1, for i = 1, ..., 4.

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Problem 3

Basic linear algebra related to eigenvalues

Let

A = eginpmatrix 1 0 0 -1 endpmatrix, and B = eginpmatrix 0 1 1 0 endpmatrix.

State if each of the following is TRUE or FALSE for the above matrices A and B. In each case, you must provide reasons/hand calculations to support your answer.

(a) The matrices A and B have same eigenvalues.

(b) Eigenvalues of AB are the product of the respective eigenvalues of A and the eigenvalues of B, that is, ?i(AB) = ?i(A)?i(B), for i = 1, 2.

(c) ?i(A + B) = ?i(A) + ?i(B), for i = 1, 2.

(d) Consider the elementwise product A ? B (this is same as the MATLAB "dot star" operation A .* B). Then ?i(A ? B) = 0, for i = 1, 2.

(e) Consider the Kronecker product A ? B = eginpmatrix a11B a12B a21B a22B endpmatrix. Then | ?i(A ? B) | = 1, for i = 1, ..., 4.

Added by Gabrielle S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Keondre Parker

Step 1

For matrix A, the characteristic equation is: $$ \det(A - \lambda I) = (1 - \lambda)(4 - \lambda) - (2)(3) = \lambda^2 - 5\lambda - 2 $$ The eigenvalues of A are the roots of this equation, which are $\lambda_1 = -1$ and $\lambda_2 = 6$. For matrix B, the Show more…

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Problem 3 Basic linear algebra related to eigenvalues Let A = (1 0; 0 -1), and B = (0 1; 1 0). State if each of the following is TRUE or FALSE for the above matrices A and B. In each case, you must provide reasons/hand calculations to support your answer. (a) The matrices A and B have same eigenvalues. (b) Eigenvalues of AB are the product of the respective eigenvalues of A and the eigenvalues of B, that is, λ_i(AB) = λ_i(A)λ_i(B), for i = 1, 2. (c) λ_i(A + B) = λ_i(A) + λ_i(B), for i = 1, 2. (d) Consider the elementwise product A ⊙ B (this is same as the MATLAB "dot star" operation A .* B). Then λ_i(A ⊙ B) = 0, for i = 1, 2. (e) Consider the Kronecker product A ⊗ B = (a_11 B a_12 B; a_21 B a_22 B). Then |λ_i(A ⊗ B)| = 1, for i = 1, ..., 4.

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