Practice Question: Diagonalization

Diagonalization

4 Activities

Consider the matrix A=[2014]A=\begin{bmatrix}2 & 0\\1&4\end{bmatrix}A=[21​04​]. Which of the following matrices PPP diagonalizes AAA?

[021−1]\begin{bmatrix}0 & 2\\1&-1\end{bmatrix}[01​2−1​]

[−1021]\begin{bmatrix}-1 & 0\\2&1\end{bmatrix}[−12​01​]

[02−11]\begin{bmatrix}0 & 2\\-1&1\end{bmatrix}[0−1​21​]

[102−1]\begin{bmatrix}1 & 0\\2&-1\end{bmatrix}[12​0−1​]

AAA is not diagonalizable.

I don't know

Diagonalization

Given that the matrix A=[−32611−1942−14−6]A=\left[\begin{array}{rrr}-3&26&11\\-1&9&4\\2&-14&-6\end{array}\right]A=​−3−12​269−14​114−6​​ has characteristic polynomial (λ+1)2(λ−2)(\lambda+1)^2(\lambda-2)(λ+1)2(λ−2) and the matrix B=[−12300−513010−30−2]B=\left[\begin{array}{rrr}-12&30&0\\-5&13&0\\10&-30&-2\end{array}\right]B=​−12−510​3013−30​00−2​​ has characteristic polynomial (λ+2)2(λ−3)(\lambda+2)^2(\lambda-3)(λ+2)2(λ−3) , determine which of A,BA,BA,B are diagonalizable, and in the case that they are diagonalizable, find the diagonal matrix DDD and the diagonalizing matrix PPP.

133 - FML 3 - 18.1W e.g. 76

 Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}}P​ and   ⁣D‾\bcb{\boldsymbol{ \ul{D}}}D​ that diagonalize the matrix   ⁣A‾=[10413−2101]\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 1 & 0 & 4 \\ 1 & 3 & -2 \\ 1 & 0 & 1 \end{bmatrix} }}A​=​111​030​4−21​​, if they exist

133 - FML 3 - 18.1W e.g. 73

 Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}}P​ and   ⁣D‾\bcb{\boldsymbol{ \ul{D}}}D​ that diagonalize the matrix   ⁣A‾=[4−162162−18]\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 4 & -1 & 6 \\ 2 & 1 & 6 \\ 2 & -1 & 8 \end{bmatrix} }}A​=​422​−11−1​668​​, if they exist
.  

133 - FML 3 - 18.1W e.g. 72

 Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}}P​ and   ⁣D‾\bcb{\boldsymbol{ \ul{D}}}D​ that diagonalize the matrix   ⁣A‾=[3−20−230005]\bcb{\boldsymbol{ \ul{A } = \begin{bmatrix} 3 & -2 & 0 \\ -2 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} }}A​=​3−20​−230​005​​, if they exist
.  

133 - FML 3 - 18.1W e.g. 68

 Find all value of k\bcb{\boldsymbol{ k}}k such that the matrix   ⁣M‾=[1k02]\bcb{\boldsymbol{ \ul{M} = \begin{bmatrix} 1 & k \\ 0 & 2 \end{bmatrix} }}M​=[10​k2​] is fully diagonalizable. 

Let
A=[−2273]A=\begin{bmatrix}-2&2\\7&3\end{bmatrix}A=[−27​23​]
Find an invertible matrix PPPand a diagonal matrix DDD such that P−1AP=DP^{-1}AP = DP−1AP=D, and thus find A10A^{10}A10.

Practice Question: Diagonalizing a Matrix

Is the following matrix diagonalizable? If so, find a diagonalization for it.
A=(−1522)A = \left ( \begin{array}{cc}
-1 & 5 \\
2  & 2
\end{array}
\right )A=(−12​52​)

Diagonalization

Practice: Diagonalization
Let A=[260−1]A=\left[\begin{array}{rr}
2&6\\
0&-1\end{array}\right]A=[20​6−1​]. 
First, convince yourself (without calculation) that AAA is diagonalizable.

Properties

Practice: Determining Diagonalizability
Let A=[−79−35]A=\left[\begin{array}{rr}-7&9\\-3&5\end{array}\right]A=[−7−3​95​]. Is AAA diagonalizable?

133 Mid 2 18.4F e.g. 15

If A‾=[−8k0−8]\A=\sm {-8}{k}{0}{-8}A​=[−80​k−8​], then determine the values of k for which A‾\AA​ is diagonalizable. 
Enter the values of k in increasing order, separated by a comma. 
For example of you find k = -2, k = 1 and k = 0, enter your answer as "-2, 0, 1". If there are none, enter the ̸ ⁣∃\not\!\exist∃ symbol (below the equation palette). 

19.4F_Final_Builder_Ch_12.7_Dynamical_Systems_$\tkcth{eg7}$_$\key{Final}$_Builder_$\tkcth{12.7.}\tkcf{7}$_

A basis for R2\mathbb{R}^2R2 is given by the set E={ξ⃗1, ξ⃗2}E = \left\{\vXi_1,\, \vXi_2 \right\}E={ξ​1​,ξ​2​}, where  ξ⃗1=[31]\vXi_1 = \colvec{3}{1}ξ​1​=[31​] and ξ⃗2=[12]\vXi_2 = \colvec{1}{2}ξ​2​=[12​] are the eigenvectors of the matrix 
A‾=[53−212]
          \A = \sm{5}{3}{-2}{12}
        A​=[5−2​312​]
Find [x⃗]E[ \vx ]_{E}[x]E​, the coordinates (in the EEE basis) of the vector whose standard coordinates are x⃗=[1−1]\vx = \colvec{1}{-1}x=[1−1​] . 

133 - FML 1 - 18.1W e.g. 10.1

A square n × n matrix AAA is said to be diagonalizable if it is similar to a diagonal matrix, i.e. A‾=P ⁣‾ D ⁣‾ P ⁣‾ −1\A = \P \bct{\D} \P^{\bct{-1}}A​=P​D​P​−1  . 
Rearranging, we have the definition that A‾\AA​ is diagonlizable if D ⁣‾ =P ⁣‾ −1A‾P ⁣‾ \D=\P^{\bct{-1}} \A \PD​=P​−1A​P​   is diagonal. 
The notation itself should make clear that this is only possible if there is an n×nn\times nn×n matrix P ⁣‾ \PP​ that is invertible (i.e. ∃  P ⁣‾ −1  :  P ⁣‾ P ⁣‾ −1  =  I ⁣‾ \exist \; \minv{\P} \; : \; \P\minv{\P} \; = \;\I∃P​−1:P​P​−1=I​). The question then arises: what is P ⁣‾ \PP​ and how can we find it?

133 - FML 1 - 18.1W e.g. 9

If B ⁣‾ = diag(λ1,λ2,λ3)\B=\text{ diag}(\lambda_1,\lambda_2,\lambda_3)B​= diag(λ1​,λ2​,λ3​) find an expression for the matrix-matrix multiplication:
 [x⃗1x⃗2x⃗3]B ⁣‾  \big[{\vx}_1\quad{\vx}_2\quad {\vx}_3\big]\B[x1​x2​x3​]B​, 
in terms of  the diagonal entries  λi\lambda_iλi​ of  B ⁣‾ \BB​,  and the 3×13 \times 13×1column vectors x⃗i\vx_ixi​. 

Practice Question: Diagonalization

Related Topics

Diagonalization

More Diagonalization Questions:

Diagonalization

133 - FML 3 - 18.1W e.g. 76

133 - FML 3 - 18.1W e.g. 73

133 - FML 3 - 18.1W e.g. 72

133 - FML 3 - 18.1W e.g. 68

Practice Question: Diagonalizing a Matrix

Diagonalization

Properties

133 Mid 2 18.4F e.g. 15

19.4F_Final_Builder_Ch_12.7_Dynamical_Systems_$\tkcth{eg7}$_$\key{Final}$_Builder_$\tkcth{12.7.}\tkcf{7}$_

133 - FML 1 - 18.1W e.g. 10.1

133 - FML 1 - 18.1W e.g. 9