High School
SAT
ACT
University
MCAT
LSAT
DAT

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

Related Topics

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

Diagonalization

4 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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.

  • If you're not sure where to start: write out the elements of the matrices and perform the multiplication, then see if you can express your answer in terms of the original terms. You may find it helpful to use a different color for each of λ1\bco{\lambda_1} , λ2\bcf{\lambda_2} and λ3\bct{\lambda_3}.

This problem is important to attempt, even if you don't figure it out on your own: it explains the entire logic of the method used to diagonalize matrices, so be sure to at least give it a shot, or the rest of the development will seem more difficult than it actually is.

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. 10.1
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 = \P \bct{\D} \P^{\bct{-1}} .
Rearranging, we have the definition that A‾\A is diagonlizable if D ⁣‾ =P ⁣‾ −1A‾P ⁣‾ \D=\P^{\bct{-1}} \A \P is diagonal.
The notation itself should make clear that this is only possible if there is an n×nn\times n matrix P ⁣‾ \P that is invertible (i.e. ∃  P ⁣‾ −1  :  P ⁣‾ P ⁣‾ −1  =  I ⁣‾ \exist \; \minv{\P} \; : \; \P\minv{\P} \; = \;\I). The question then arises: what is P ⁣‾ \P and how can we find it?
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?