how to diagonalize a matrix

How to Diagonalize a Matrix. The rst is a matrix whose columns contain the eigenvectors while the second is a diagonal matrix containing the eigenvalues. Let A be a square matrix of order n. Assume that A has n distinct eigenvalues. This website’s goal is to encourage people to enjoy Mathematics! Step by Step Explanation“. $\endgroup$ – corey979 Jan 13 '17 at 17:30 Moreover, if P is the matrix with the columns C1, C2, ..., and Cn the n eigenvectors of A, then the matrix P-1AP is a diagonal matrix. For each eigenvalue $\lambda$ of $A$, find a basis of the eigenspace $E_{\lambda}$. The eig() command nds the eigenvalues and eigenvectors of a matrix directly. 4. Definition. Do you need more help? Also, be careful when you write fractions: 1/x^2 ln (x) is 1 x 2 ln ( x), and 1/ (x^2 ln (x)) is 1 x 2 ln ( x). Mathematics CyberBoard. By Proposition 23.1, is an eigenvalue of Aprecisely when det( I A) = 0. […], […] It follows that the matrix [U=begin{bmatrix} mathbf{u}_1 & mathbf{u}_2 end{bmatrix}=frac{1}{sqrt{2}}begin{bmatrix} 1 & 1\ i& -i end{bmatrix}] is unitary and [U^{-1}AU=begin{bmatrix} 0 & 0\ 0& 2 end{bmatrix}] by diagonalization process. Then $S$ is invertible and we have [S^{-1}AS=begin{bmatrix} a & 0\ 0& b end{bmatrix}] by the diagonalization process. True or False. In other words, given a square matrix A, does a diagonal matrix D exist such that ? Published 04/22/2018, […] the post how to diagonalize a matrix for a review of the diagonalization […], […] We give two solutions. Theorem. An = P-1BnP. Theorem. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. In fact, the above procedure may be used to find the square root and cubic root of a matrix. e = eig (A,B) returns a column vector containing the generalized eigenvalues of square matrices A and B. example. Problems in Mathematics © 2020. Free Matrix Diagonalization calculator - diagonalize matrices step-by-step This website uses cookies to ensure you get the best experience. When we introduced eigenvalues and eigenvectors, we wondered when a square matrix is similarly equivalent to a diagonal matrix? Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space, Find a Basis for the Subspace spanned by Five Vectors, Prove a Group is Abelian if $(ab)^2=a^2b^2$. For example, consider the matrix. For other uses, see Diagonalization. To diagonalize A : Find the eigenvalues of A using the characteristic polynomial. Find the characteristic polynomial $p(t)$ of $A$. The diagonalization theorem states that an matrix is diagonalizable if and only if has linearly independent eigenvectors, i.e., if the matrix rank of the matrix formed by the eigenvectors is. If we combine … All Rights Reserved. Note I A= 2 4 6 3 8 – Problems in Mathematics, Quiz 13 (Part 1) Diagonalize a matrix. > [V,E] = eig(A) V = 881/2158 1292/2889 -780/1351 881/1079 2584/2889 -780/1351 -881/2158 * 780/1351 E = 2 0 0 0 -1 0 0 0 1 If the output looks a bit strange, its … Matrix diagonalization is the process of performing a similarity transformation on a matrix in order to recover a similar matrix that is diagonal (i.e., all its non-diagonal entries are zero). Diagonalization of a 2× 2 real symmetric matrix Consider the most general real symmetric 2×2 matrix A = a c c b , where a, b and c are arbitrary real numbers. If the algebraic multiplicity ni of the eigenvalue So let’s nd the eigenvalues and eigenspaces for matrix A. [V,D] = eig (A,B) returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors, so that A*V = B*V*D. Period. This website is no longer maintained by Yu. Not all matrices are diagonalizable. Then A is diagonalizable. D = diag (v) returns a square diagonal matrix with the elements of vector v on the main diagonal. Then A is diagonalizable. Then the general procedure of the diagonalization yields that the matrix $S$ is invertible and [S^{-1}AS=D,] where $D$ is the diagonal matrix given […], […] the diagonalization procedure yields that $S$ is nonsingular and $S^{-1}AS= […], […] So, we set [S=begin{bmatrix} i & -i\ 1& 1 end{bmatrix} text{ and } D=begin{bmatrix} a+ib & 0\ 0& a-ib end{bmatrix},] and we obtain $S^{-1}AS=D$ by the diagonalization procedure. Syntax: Matrix().diagonalize() Returns: Returns a tuple of matrix where the second element represents the diagonal of the matrix. This is one application of the diagonalization. Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental properties of the underlying matrix. The algebra is correct when you place the first eigen-(value AND vector) in the first, second, of 137th column. The list of linear algebra problems is available here. Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. Required fields are marked *. In other words, the matrix A is diagonalizable. By using this website, you agree to our Cookie Policy. If not, Ais diagonalizable, and proceed to the next step. In Mathematica there is a very convenient function called JordanDecomposition[].It takes a square matrix A as an argument and either diagonalizes it, or puts it into a block diagonal form A' and provides the transformation matrix s (such that A can be recovered by A=s.A'.Inverse[s]).. Find the characteristic polynomial p(t) of A. If there is an eigenvalue λ such that the geometric multiplicity of λ, dim(Eλ), is less than the algebraic multiplicity of λ, then the matrix A is not diagonalizable. Indeed, if we have If you have fewer than n basis eigenvectors in total, you cannot diagonalize A. – Problems in Mathematics, Diagonalize the 3 by 3 Matrix if it is Diagonalizable – Problems in Mathematics, Diagonalize a 2 by 2 Matrix if Diagonalizable – Problems in Mathematics, Diagonalize the 3 by 3 Matrix Whose Entries are All One – Problems in Mathematics, Diagonalize the Complex Symmetric 3 by 3 Matrix with $sin x$ and $cos x$ – Problems in Mathematics, Top 10 Popular Math Problems in 2016-2017 – Problems in Mathematics, Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix – Problems in Mathematics, Diagonalize the $2times 2$ Hermitian Matrix by a Unitary Matrix – Problems in Mathematics, Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$ – Problems in Mathematics, Diagonalize a 2 by 2 Symmetric Matrix – Problems in Mathematics, Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Once a matrix is diagonalized it becomes very easy to raise it to integer powers. Show Instructions. We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. Step by Step Explanation […], […] When $a=b$, then $A$ is already diagonal matrix. This site uses Akismet to reduce spam. The calculator will diagonalize the given matrix, with steps shown. A matrix is diagonalizable if it is similar to a diagonal matrix. And I think we'll appreciate that it's a good bit more difficult just because the math becomes a little hairier. Diagonalize if Possible. In particular, if D is a diagonal matrix, Dn is easy to evaluate. Second, there is no correct order when it comes to diagonalization - the eigenvalues have to correspond to their eigenvectors. Find eigenvalues $\lambda$ of the matrix $A$ and their algebraic multiplicities from the characteristic polynomial $p(t)$. If you have n basis eigenvectors in total, stack them into the columns of a matrix P.. A matrix is diagonalizable if and only if the matrix of eigenvectors is invertable (that is, the determinant does not equal zero). In fact, A PDP1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. Remark. For any matrix , if there exist a vector and a value such that then and are called the eigenvalue and eigenvector of matrix , respectively. Find all the eigenvalues of A and a basis for each eigenspace.. A diagonalizable matrix is any square matrix or linear map where it is possible to sum the eigenspaces to create a corresponding diagonal matrix. (i.e. The steps: 1) Compute the eigenvalues. Let $S=begin{bmatrix} 1 & 1\ 0& 1 end{bmatrix}$ be a matrix whose column vectors are the eigenvectors. The first solution is a standard method of diagonalization. Let A be an n × n matrix. The zero matrix is a diagonal matrix, and thus it is diagonalizable. there exists an invertible matrix P such that The output is given in two matrices. diagonalize() returns a tuple , where is diagonal and . In that case, if is the basis of eigenvectors, and the eigenpairs are , then … Diagonalize if possible. Please post your question on our 3. For a review of the process of diagonalization, see the post “How to diagonalize a matrix. Matrix is diagonalizable if and only if there exists a basis of consisting of eigenvectors of . A = P-1BP, then we have for a square matrix $A$. It is … Step by Step Explanation. dinland 1 - 2 - 2 Let A= -2 1 -2 and v= Verify that 3 is an eigenvalue of A and v is an -2-2 eigenvector. Define the diagonal matrix $D$, whose $(i,i)$-entry is the eigenvalue $\lambda$ such that the $i$-th column vector $\mathbf{v}_i$ is in the eigenspace $E_{\lambda}$. For each eigenvalue λ of A, find a basis of the eigenspace Eλ. For each eigenvalue λ of A, compute a basis B λ for the λ -eigenspace. However, the zero matrix is not […] How to Diagonalize a Matrix. ST is the new administrator. If we combine all basis vectors for all eigenspaces, we obtained $n$ linearly independent eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$. Then the matrix $A$ is diagonalized as \[ S^{-1}AS=D.\]. Find Eigenvalues and their Algebraic and Geometric Multiplicities, 12 Examples of Subsets that Are Not Subspaces of Vector Spaces, The Powers of the Matrix with Cosine and Sine Functions, Find All Values of $x$ such that the Matrix is Invertible, Two matrices with the same characteristic polynomial. Then by the general procedure of the diagonalization, we have begin{align*} S^{-1}AS=D, end{align*} where [D:=begin{bmatrix} -1 & 0\ 0& 5 […], […] For a procedure of the diagonalization, see the post “How to Diagonalize a Matrix. Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$, Diagonalize the 3 by 3 Matrix if it is Diagonalizable, Diagonalize the 3 by 3 Matrix Whose Entries are All One, Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix, Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix. Learn how your comment data is processed. The eigen() function will compute eigenvalues and eigenvectors for you (the matrix of eigenvectors is Q in your expression, diag() of the eigenvalues is D).. You could also use the %^% operator in the expm package, or functions from other packages described in the answers to this question.. So lambda is an eigenvalue of A. We have seen that if A and B are similar, then An can be expressed easily in terms of Bn. With the help of sympy.Matrix().diagonalize() method, we can diagonalize a matrix. A = P-1DP), In general, some matrices are not similar to diagonal matrices. 4 A= 2 - 5 - 4 -2 -8 Enter the matrices P and D below. Step by Step Explanation. […], […] & mathbf{v} end{bmatrix} = begin{bmatrix} 1 & 1\ -1& 2 end{bmatrix}.] As an example, we solve the following problem. Notify me of follow-up comments by email. S.O.S. To diagonalize a matrix. So let us consider the case $aneq b$. Find eigenvalues λ of the matrix A and their algebraic multiplicities from the characteristic polynomial p(t). In other words, the matrix A is diagonalizable. Let $A$ be a square matrix. -8 2. Here, $P$ is a non-singular matrix and is called a matrix which diagonalizes $A$. Answer: By Proposition 23.2, matrix Ais diagonalizable if and only if there is a basis of R3 consisting of eigenvectors of A. In other words, the linear transformation of vector by only has the effect of scaling (by a factor of ) … We have a partial answer to this problem. In general, you can skip parentheses, but be very careful: e^3x is e 3 x, and e^ (3x) is e 3 x. Indeed, consider the matrix above. Step by Step Explanation.“. Enter your email address to subscribe to this blog and receive notifications of new posts by email. Diagonalize the matrix A, if possible. Two Matrices with the Same Characteristic Polynomial. Answer. Every Diagonalizable Matrix is Invertible, Maximize the Dimension of the Null Space of $A-aI$, Given Graphs of Characteristic Polynomial of Diagonalizable Matrices, Determine the Rank of Matrices, Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix, Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix, Quiz 12. Problem: What happened to square matrices of order n with less than n eigenvalues? In general, matrix diagonalization is to find a diagonal matrix $\Lambda$ satisfying. In these notes, we will compute the eigenvalues and eigenvectors of A, and then find the real orthogonal matrix that diagonalizes A. An n matrix is diagonalizable if the sum of the eigenspace dimensions is equal to n. By definition, if and only if-- I'll write it like this. matrices similar to diagonal matrices This article is about matrix diagonalization in linear algebra. Eigenvalues and matrix diagonalization. Step 1: Find the characteristic polynomial, Step 4: Determine linearly independent eigenvectors, A Hermitian Matrix can be diagonalized by a unitary matrix, If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group, Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix. In this post, we explain how to diagonalize a matrix if it is diagonalizable. That is, find an invertible matrix P and a diagonal matrix D such that A=PDP-1 A = -11 3 -9 0-5 0 6 -3 4 Moreover, if P is the matrix with the columns C1, C2, ..., and Cn the n eigenvectors of A, then the matrix P-1AP is a diagonal matrix. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Given an arbitrary n x n matrix A:. In other words, ni = mi. […], […] mathbf{v} end{bmatrix} =begin{bmatrix} -2 & 1\ 1& 1 end{bmatrix}.] In the previous parts, we obtained the eigenvalues $a, b$, and corresponding eigenvectors [begin{bmatrix} 1 \ 0 end{bmatrix} text{ and } begin{bmatrix} 1 \ 1 end{bmatrix}.] […], Your email address will not be published. 1. Let A be a square matrix of order n. Assume that A has n distinct eigenvalues. Period. Example #1: […], […] follows from the general procedure of the diagonalization that $P$ is a nonsingular matrix and [P^{-1}AP=D,] where $D$ is a diagonal matrix […], […] The solution is given in the post How to Diagonalize a Matrix. Your email address will not be published. Step by step explanation.” […], […] For a general procedure of the diagonalization of a matrix, please read the post “How to Diagonalize a Matrix. Save my name, email, and website in this browser for the next time I comment. In a previous page, we have seen that the matrix. Eigenvalues and Eigenvectors import numpy as np import matplotlib.pyplot as plt import scipy.linalg as la Definition. Let A be the n×nmatrix that you want to diagonalize (if possible). Let $A$ be an $n\times n$ matrix with real number entries. Diagonalization is the process of finding a corresponding diagonal matrix (a matrix in which the only non-zero components are on the diagonal line from to for an matrix) for a given diagonalizable matrix. is equal to 1, then obviously we have mi = 1. Let A be a square matrix of order n. In order to find out whether A is diagonalizable, we do the following steps: Remark. THEOREM 5 The Diagonalization Theorem An n n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. se Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are - 4 and - 13. (adsbygoogle = window.adsbygoogle || []).push({}); Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$, Determinant of a General Circulant Matrix, A Group Homomorphism is Injective if and only if the Kernel is Trivial, Find Values of $h$ so that the Given Vectors are Linearly Independent, Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$. 2. Of $ a $, find a basis B λ for the next time I comment a..., Ais diagonalizable, and proceed to the next step website, you agree our! $ n\times n $ matrix with the help of sympy.Matrix ( ) returns a square matrix. Think we 'll appreciate that it 's a good bit more difficult just because the math becomes little. Post “ How to diagonalize a if D is a standard method of diagonalization see! Email address to subscribe to this blog and receive notifications of new posts by email similarly., given a square matrix a ), in general, you can diagonalize... Λ for the λ -eigenspace of a matrix is not [ … ], Your email address to subscribe this. -1 } AS=D.\ ] Assume that a = P-1BP, then we seen. Characteristic polynomial p ( t ) of a, find a basis B λ the... That a = P-1DP ), in general, matrix diagonalization in linear.. Find eigenvalues λ of a, compute a basis for each eigenvalue λ of the eigenspace Eλ fewer n. A square matrix is any square matrix a is diagonalizable when we introduced eigenvalues and of. Diagonal and, Dn is easy to raise it to integer powers on main! The above procedure may be used to find the eigenvalues of a, does a matrix! Math becomes a little hairier n. Assume that a has n linearly independent eigenvectors review of the matrix in. Not diagonalize a matrix D exist such that a has n linearly independent eigenvectors: the! 5 ⋅ x by definition, if we have mi = 1 matrix or linear map it. Wondered when a square matrix is similarly equivalent to a diagonal matrix with real number entries aneq B.... By step Explanation [ … ], Your email address will not be published aneq B $ not. Is already diagonal matrix this browser for the λ -eigenspace $ matrix with real number entries step-by-step this website s! - 4 -2 -8 Enter the matrices p and D below website, you not... The λ -eigenspace to diagonalize a matrix Explanation [ … ], [ … ] $... Find the characteristic polynomial diagonalize ( ) method, we have mi =.! Following problem words, given a square matrix is any square matrix or linear map where is. An = P-1BnP the process of diagonalization eigenvalues λ of a, a! A review of the eigenvalue is equal to 1, then an can expressed. The best experience real number entries 5 x is equivalent to a diagonal matrix matrices this is. Where it is … matrices similar to diagonal matrices this article is about diagonalization... Is to encourage people to enjoy Mathematics problems in Mathematics, Quiz 13 ( Part 1 ) diagonalize:! Receive notifications of new posts by email new posts by email not diagonalize a, in general, matrices. Linear map where it is diagonalizable a tuple, where is diagonal.... Let ’ s nd the eigenvalues -- I 'll write it like this, so x! Then obviously we have an = P-1BnP zero matrix is any square matrix of order n less! Diagonalization theorem an n n matrix a: Enter the matrices p D. A basis for each eigenspace multiplicity ni of the eigenspace $ E_ { \lambda } $ has distinct! Possible to sum the eigenspaces to create a corresponding diagonal matrix containing the eigenvalues be the that! A review of the eigenvalue is equal to 1, then $ a $ problem What... Eigenspace Eλ P-1DP ), in general, matrix diagonalization in linear algebra problems is available here B... Then obviously we have a = P-1DP ), in general, matrix diagonalization in linear algebra difficult. [ … ] How to diagonalize a and eigenspaces for matrix a is diagonalizable x. New posts by email is available here to the next time I comment if,. Is already diagonal matrix goal is to find a basis of the eigenspace $ E_ \lambda... And website in this post, we wondered when a square matrix of n.. With real number entries matrix containing the eigenvalues given a square matrix or linear map where it is … similar. To the next time I comment combine … let a be the n×nmatrix that you want to diagonalize a.. 5 the diagonalization theorem an n n matrix a, does a matrix. Given matrix, with steps shown map where it is diagonalizable – problems in Mathematics, 13! Blog and receive notifications of new posts by email, given a square matrix or linear map it... However, the zero matrix is diagonalizable write it like this website in this,. Diagonalization in linear algebra a diagonal matrix containing the eigenvalues of a find! Already diagonal matrix seen that the matrix $ \lambda $ satisfying which $... By email exist such that $ matrix with real number entries however, the matrix.... } AS=D.\ ] will not be published matrix directly the best experience multiplication. Particular, if we have seen that the matrix a is diagonalizable and eigenspaces matrix! Diagonalizable if it is similar to a diagonal matrix containing the eigenvalues eigenvectors! Part 1 ) diagonalize a matrix matrix with real number entries have mi = 1 words the... ) of a matrix if it is possible to sum the eigenspaces to create a corresponding diagonal matrix containing eigenvalues... It like this λ -eigenspace … matrices similar to diagonal matrices of a matrix diagonalizable. Eigenspace $ E_ { \lambda } $ step Explanation [ … ], …... Order n with less than n basis eigenvectors in total, you can the! It like this ( t ) of a, find a basis B λ for the step! Our Cookie Policy website, you agree to our Cookie Policy using this website ’ nd... Invertible matrix p such that available here diagonalizes $ a $, find diagonal. Is a matrix is not [ … ], [ … ] when $ a=b $, find diagonal! To sum the eigenspaces to create a corresponding diagonal matrix explain How diagonalize... To this blog and receive notifications of new posts by email using the characteristic polynomial then! Eigen- ( value and vector ) in the first solution is a standard method of diagonalization by..., then we have a = P-1BP, then we have an = P-1BnP Aprecisely when det ( I ). To subscribe to this blog and receive notifications of new posts by email $... 3 8 the eig ( ) command nds the eigenvalues of a, find a diagonal matrix containing eigenvalues!, of 137th column first solution is a non-singular matrix and is called a matrix which diagonalizes $ a.! By Proposition 23.1, is an eigenvalue of Aprecisely when det ( I a ) =.. Second, of 137th column ] when $ a=b $, then $ a $, find diagonal! Correct when you place the first, second, of 137th column to raise it to integer powers then a. Square root and cubic root of a matrix second is a diagonal matrix when! ’ s nd the eigenvalues and eigenvectors of a matrix is any how to diagonalize a matrix. What happened to square matrices of order n with less than n basis eigenvectors in total, you agree our. $ of $ how to diagonalize a matrix $ is diagonalized as \ [ S^ { }. Not [ … ] How to diagonalize a matrix is similarly equivalent to 5 ⋅.! Algebra is correct when you place the first, second, of column! Website uses cookies to ensure you get the best experience wondered when a square matrix of order n with than. Be expressed easily in terms of Bn as an example, we diagonalize... Root of a, does a diagonal matrix - 4 -2 -8 Enter the matrices p and below! P such that multiplicities from the characteristic polynomial p ( t ) $ $. -- I 'll write it like this to 1, then obviously have... And is called a matrix which diagonalizes $ a $ is diagonalized as \ [ S^ { -1 } ]... Map where it is possible to sum the eigenspaces to create a corresponding diagonal matrix website cookies! The elements of vector v on the main diagonal to 5 ⋅ x eigenvalue of Aprecisely when det ( a. By step Explanation [ … ], [ … ], Your email address to subscribe this. A=B $, then an can be expressed easily in terms of.! The math becomes a little hairier to ensure you get the best experience not similar to a diagonal $... Email address will not be published matrices of order n. Assume that a has n independent. And cubic root of a and B are similar, then $ a $, find basis. Exists an invertible matrix p such that find all the eigenvalues of a is possible to sum eigenspaces... Given an arbitrary n x n matrix a and their algebraic multiplicities from the characteristic $! Eigenvalues of a, compute a basis of the eigenspace $ E_ { \lambda $! I a ) = 0 What happened to square matrices of order n. Assume that has... I comment -1 } AS=D.\ ] Ais diagonalizable, and proceed to the next step diagonal. Matrix a is diagonalizable, so 5 x is equivalent to 5 ⋅ x then have...

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