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

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Wize University Linear Algebra Textbook > Eigenvalues and Eigenvectors (Spectral Theory)

Diagonalization

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A square n × n matrix AA is said to be diagonalizable if it is similar to a diagonal matrix, i.e. A‾=P ⁣‾ D ⁣‾ P ⁣‾ −1\A = \matP \bct{\D} \matP^{\bct{-1}} .

Rearranging, we have the definition that A‾\A is diagonlizable if D ⁣‾ =P ⁣‾ −1A‾P ⁣‾ \D=\matP^{\bct{-1}} \A \matP is diagonal.

The notation itself should make clear that this is only possible if there is an n×nn\times n matrix P ⁣‾ \matP that is invertible (i.e. ∃  P ⁣‾ −1  :  P ⁣‾ P ⁣‾ −1  =  I ⁣‾ \exist \; \minv{\matP} \; : \; \matP\minv{\matP} \; = \;\I). The question then arises: what is P ⁣‾ \matP and how can we find it?


  • e.g. For an n×nn\times n matrix A‾\A, assume P ⁣‾ \matP is an n×nn\times n invertible matrix, and D ⁣‾ = diag(λ1,λ2,λ3,…,λn)\D = \text{ diag}(\lambda_1 , \lambda_2 , \lambda_3 , \ldots, \lambda_n ). Without assuming anything else, use the definition of diagonalizability (viz. A‾=P ⁣‾ D ⁣‾ P ⁣‾ −1\A = \matP \D \minv{\matP} ) to find a necessary and sufficient condition for defining the matrix P ⁣‾ \matP by completing the following steps:

  1. Right-multiply the definition for diagonalizability by P ⁣‾ \matP:
  2. Let x⃗i{\vx}_{\bco{i}} be the ith\bco{i}^{th} column of the matrix P ⁣‾ \matP, then P ⁣‾ =\matP=
  3. Using your understanding of the nature of matrix multiplication in general, as well as the special case of right-multiplication of a matrix by a diagonal matrix, use the above representation of P ⁣‾ \matP to find column representations of A‾P ⁣‾ \A\matP and P ⁣‾ D ⁣‾ \matP \D.






More Diagonalization Questions:
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] has characteristic polynomial (λ+1)2(λ−2)(\lambda+1)^2(\lambda-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] has characteristic polynomial (λ+2)2(λ−3)(\lambda+2)^2(\lambda-3) , determine which of A,BA,B are diagonalizable, and in the case that they are diagonalizable, find the diagonal matrix DD and the diagonalizing matrix PP.
133 - FML 3 - 18.1W e.g. 76
Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}} and   ⁣D‾\bcb{\boldsymbol{ \ul{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} }}, if they exist
133 - FML 3 - 18.1W e.g. 73
Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}} and   ⁣D‾\bcb{\boldsymbol{ \ul{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} }}, if they exist
.
133 - FML 3 - 18.1W e.g. 72
Find the matrices   ⁣P‾\bcb{\boldsymbol{ \ul{P}}} and   ⁣D‾\bcb{\boldsymbol{ \ul{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} }}, if they exist
.
133 - FML 3 - 18.1W e.g. 68
Find all value of k\bcb{\boldsymbol{ k}} such that the matrix   ⁣M‾=[1k02]\bcb{\boldsymbol{ \ul{M} = \begin{bmatrix} 1 & k \\ 0 & 2 \end{bmatrix} }} is fully diagonalizable.
Let
A=[−2273]A=\begin{bmatrix}-2&2\\7&3\end{bmatrix}
Find an invertible matrix PPand a diagonal matrix DD such that P−1AP=DP^{-1}AP = D, and thus find A10A^{10}.
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 )
Diagonalization
Practice: Diagonalization
Let A=[260−1]A=\left[\begin{array}{rr} 2&6\\ 0&-1\end{array}\right].
First, convince yourself (without calculation) that AA is diagonalizable.
Properties
Practice: Determining Diagonalizability
Let A=[−79−35]A=\left[\begin{array}{rr}-7&9\\-3&5\end{array}\right]. Is AA diagonalizable?
133 Mid 2 18.4F e.g. 15
If A‾=[−8k0−8]\A=\sm {-8}{k}{0}{-8}, then determine the values of k for which A‾\A 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}^2 is given by the set E={ξ⃗1, ξ⃗2}E = \left\{\vXi_1,\, \vXi_2 \right\}, where ξ⃗1=[31]\vXi_1 = \colvec{3}{1} and ξ⃗2=[12]\vXi_2 = \colvec{1}{2} are the eigenvectors of the matrix
A‾=[53−212] \A = \sm{5}{3}{-2}{12}
Find [x⃗]E[ \vx ]_{E}, the coordinates (in the EE basis) of the vector whose standard coordinates are x⃗=[1−1]\vx = \colvec{1}{-1} .
133 - FML 1 - 18.1W e.g. 9
If B ⁣‾ = diag(λ1,λ2,λ3)\B=\text{ diag}(\lambda_1,\lambda_2,\lambda_3) find an expression for the matrix-matrix multiplication:
[x⃗1x⃗2x⃗3]B ⁣‾  \big[{\vx}_1\quad{\vx}_2\quad {\vx}_3\big]\B,
in terms of the diagonal entries λi\lambda_i of B ⁣‾ \B, and the 3×13 \times 1column vectors x⃗i\vx_i.
Practice Question: Diagonalization
Consider the matrix A=[2014]A=\begin{bmatrix}2 & 0\\1&4\end{bmatrix}. Which of the following matrices PP diagonalizes AA?