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Question 4. We study the diagonalization of 2 × 2 matrices with a 0 in the upper right corner $A = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$. (a) Give a proof of the statement “ if ? is an eigenvalue of A, then there is $\vec{v} \neq \vec{0}$ such that $A\vec{v} = \lambda\vec{v}$”. (b) Give a proof of the statement “if $\lambda_1 \neq \lambda_2$ and $\vec{v}_1, \vec{v}_2$ are eigenvectors with eigenvalues $\lambda_1, \lambda_2$ respectively, then $\vec{v}_1, \vec{v}_2$ are linearly independent. (c) Show that if $a \neq c$, then the matrix $A = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$ is diagonalizable. (d) Give conditions on $a, b$ that are equivalent to the matrix $A = \begin{bmatrix} a & 0 \\ b & a \end{bmatrix}$ being diagonaliz- able.

          Question 4. We study the diagonalization of 2 × 2 matrices with a 0 in the upper right
corner $A = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$.
(a) Give a proof of the statement “ if ? is an eigenvalue of A, then there is $\vec{v} \neq \vec{0}$ such that
$A\vec{v} = \lambda\vec{v}$”.
(b) Give a proof of the statement “if $\lambda_1 \neq \lambda_2$ and $\vec{v}_1, \vec{v}_2$ are eigenvectors with eigenvalues
$\lambda_1, \lambda_2$ respectively, then $\vec{v}_1, \vec{v}_2$ are linearly independent.
(c) Show that if $a \neq c$, then the matrix $A = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}$ is diagonalizable.
(d) Give conditions on $a, b$ that are equivalent to the matrix $A = \begin{bmatrix} a & 0 \\ b & a \end{bmatrix}$ being diagonaliz-
able.

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Question 4. We study the diagonalization of 2 × 2 matrices with a 0 in the upper right
corner A =
< b m a t r i x >.
(a) Give a proof of the statement “ if ? is an eigenvalue of A, then there is v⃗≠0⃗ such that
Av⃗ = λv⃗”.
(b) Give a proof of the statement “if λ1 ≠λ2 and v⃗1, v⃗2 are eigenvectors with eigenvalues
λ1, λ2 respectively, then v⃗1, v⃗2 are linearly independent.
(c) Show that if a ≠ c, then the matrix A =
< b m a t r i x > is diagonalizable.
(d) Give conditions on a, b that are equivalent to the matrix A =
< b m a t r i x > being diagonaliz-
able.

Added by Donald P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

08/31/2023

Step 1

This means there exists a non-zero vector v such that Av = λv. Let v = [x, y]. Then, Av = [[a, 0], [b, c]] * [x, y] = [ax, by]. Since Av = λv, we have [ax, by] = λ[x, y]. This gives us the system of equations: ax = λx by = λy From the first equation, we get x = 0 Show more…

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Question 4. We study the diagonalization of 2 imes 2 matrices with a 0 in the upper right corner A=[[a,0],[b,c]]. (a) Give a proof of the statement "if lambda is an eigenvalue of A, then there is vec(v)!=vec(0) such that Avec(v)=lambda vec(v). (b) Give a proof of the statement " if lambda _(1)!=lambda _(2) and vec(v)_(1),vec(v)_(2) are eigenvectors with eigenvalues lambda _(1),lambda _(2) respectively, then vec(v)_(1),vec(v)_(2) are linearly independent. (c) Show that if a!=c, then the matrix A=[[a,0],[b,c]] is diagonalizable. (d) Give conditions on a,b that are equivalent to the matrix A=[[a,0],[b,a]]. being diagonalizable. Question 4. We study the diagonalization of 2 2 matrices with a 0 in the upper right corner A [a0 bc] (a)Give a proof of the statement " if is an eigenvalue of A, then there is 0 such that A=X b) Give a proof of the statement if and 1, are eigenvectors with eigenvalues XX respectively,then , are linearly independent. (c) Show that if a c,then the matrix A = a0 bc] is diagonalizable able.

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