A diagonalizable matrix is any square matrix or linear map where it is possible to sum the eigenspaces to create a corresponding diagonal matrix. Then the matrix $A$ is diagonalized as $S^{-1}AS=D.$. Matrix is diagonalizable if and only if there exists a basis of consisting of eigenvectors of . In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x. $\endgroup$ – corey979 Jan 13 '17 at 17:30 2. Free Matrix Diagonalization calculator - diagonalize matrices step-by-step This website uses cookies to ensure you get the best experience. Step by Step Explanation. Period. Find eigenvalues $\lambda$ of the matrix $A$ and their algebraic multiplicities from the characteristic polynomial $p(t)$. – Problems in Mathematics, Quiz 13 (Part 1) Diagonalize a matrix. Please post your question on our The eig() command nds the eigenvalues and eigenvectors of a matrix directly. In other words, the matrix A is diagonalizable. This is one application of the diagonalization. In particular, if D is a diagonal matrix, Dn is easy to evaluate. […], […] & mathbf{v} end{bmatrix} = begin{bmatrix} 1 & 1\ -1& 2 end{bmatrix}.] Answer: By Proposition 23.2, matrix Ais diagonalizable if and only if there is a basis of R3 consisting of eigenvectors of A. > [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 … 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$by the diagonalization procedure. e = eig (A,B) returns a column vector containing the generalized eigenvalues of square matrices A and B. example. Step by Step Explanation […], […] When$a=b$, then$A$is already diagonal matrix. Enter your email address to subscribe to this blog and receive notifications of new posts by email. Step by Step Explanation.“. So lambda is an eigenvalue of A. Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental properties of the underlying matrix. THEOREM 5 The Diagonalization Theorem An n n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. there exists an invertible matrix P such that 3. Step 1: Find the characteristic polynomial, Step 4: Determine linearly independent eigenvectors, A Hermitian Matrix can be diagonalized by a unitary matrix, If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group, Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix. […], […] follows from the general procedure of the diagonalization that$P$is a nonsingular matrix and [P^{-1}AP=D,] where$D$is a diagonal matrix […], […] The solution is given in the post How to Diagonalize a Matrix. Definition. for a square matrix$A$. In other words, the matrix A is diagonalizable. Then A is diagonalizable. Let A be the n×nmatrix that you want to diagonalize (if possible). To diagonalize A : Find the eigenvalues of A using the characteristic polynomial. That is, find an invertible matrix P and a diagonal matrix D such that A=PDP-1 A = -11 3 -9 0-5 0 6 -3 4 Two Matrices with the Same Characteristic Polynomial. The calculator will diagonalize the given matrix, with steps shown. Learn how your comment data is processed. Problem: What happened to square matrices of order n with less than n eigenvalues? Example #1: [V,D] = eig (A,B) returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors, so that A*V = B*V*D. Let A be a square matrix of order n. Assume that A has n distinct eigenvalues. (i.e. If we combine … In these notes, we will compute the eigenvalues and eigenvectors of A, and then ﬁnd the real orthogonal matrix that diagonalizes A. The zero matrix is a diagonal matrix, and thus it is diagonalizable. Here,$P$is a non-singular matrix and is called a matrix which diagonalizes$A$. If we combine all basis vectors for all eigenspaces, we obtained$n$linearly independent eigenvectors$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$. Mathematics CyberBoard. Eigenvalues and matrix diagonalization. In a previous page, we have seen that the matrix. An = P-1BnP. All Rights Reserved. If there is an eigenvalue λ such that the geometric multiplicity of λ, dim(Eλ), is less than the algebraic multiplicity of λ, then the matrix A is not diagonalizable. In general, you can skip parentheses, but be very careful: e^3x is e 3 x, and e^ (3x) is e 3 x. A = P-1BP, then we have The algebra is correct when you place the first eigen-(value AND vector) in the first, second, of 137th column. se Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are - 4 and - 13. […], […] It follows that the matrix [U=begin{bmatrix} mathbf{u}_1 & mathbf{u}_2 end{bmatrix}=frac{1}{sqrt{2}}begin{bmatrix} 1 & 1\ i& -i end{bmatrix}] is unitary and [U^{-1}AU=begin{bmatrix} 0 & 0\ 0& 2 end{bmatrix}] by diagonalization process. If the algebraic multiplicity ni of the eigenvalue For each eigenvalue λ of A, find a basis of the eigenspace Eλ. A = P-1DP), In general, some matrices are not similar to diagonal matrices. Remark. Do you need more help? The rst is a matrix whose columns contain the eigenvectors while the second is a diagonal matrix containing the eigenvalues. As an example, we solve the following problem. Let A be an n × n matrix. Indeed, consider the matrix above. The output is given in two matrices. Let A be a square matrix of order n. In order to find out whether A is diagonalizable, we do the following steps: Remark. For example, consider the matrix. Range, Null Space, Rank, and Nullity of a Linear Transformation from$\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$. Notify me of follow-up comments by email. Find all the eigenvalues of A and a basis for each eigenspace.. Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. Every Diagonalizable Matrix is Invertible, Maximize the Dimension of the Null Space of$A-aI$, Given Graphs of Characteristic Polynomial of Diagonalizable Matrices, Determine the Rank of Matrices, Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix, Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix, Quiz 12. Theorem. Let$S=begin{bmatrix} 1 & 1\ 0& 1 end{bmatrix}be a matrix whose column vectors are the eigenvectors. It is … For each eigenvalue λ of A, compute a basis B λ for the λ -eigenspace. 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. LetA$be a square matrix. To diagonalize a matrix. -8 2. 1. In Mathematica there is a very convenient function called JordanDecomposition[].It takes a square matrix A as an argument and either diagonalizes it, or puts it into a block diagonal form A' and provides the transformation matrix s (such that A can be recovered by A=s.A'.Inverse[s]).. Theorem. Find the characteristic polynomial$p(t)$of$A$. Required fields are marked *. Find eigenvalues λ of the matrix A and their algebraic multiplicities from the characteristic polynomial p(t). And I think we'll appreciate that it's a good bit more difficult just because the math becomes a little hairier. Moreover, if P is the matrix with the columns C1, C2, ..., and Cn the n eigenvectors of A, then the matrix P-1AP is a diagonal matrix. In general, matrix diagonalization is to find a diagonal matrix$\Lambda$satisfying. An n matrix is diagonalizable if the sum of the eigenspace dimensions is equal to n. So let’s nd the eigenvalues and eigenspaces for matrix A. matrices similar to diagonal matrices This article is about matrix diagonalization in linear algebra. However, the zero matrix is not […] How to Diagonalize a Matrix. The diagonalization theorem states that an matrix is diagonalizable if and only if has linearly independent eigenvectors, i.e., if the matrix rank of the matrix formed by the eigenvectors is. In this post, we explain how to diagonalize a matrix if it is diagonalizable. Once a matrix is diagonalized it becomes very easy to raise it to integer powers. This site uses Akismet to reduce spam. (adsbygoogle = window.adsbygoogle || []).push({}); Non-Example of a Subspace in 3-dimensional Vector Space$\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$. […], Your email address will not be published. ST is the new administrator. We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. In fact, the above procedure may be used to find the square root and cubic root of a matrix. This website’s goal is to encourage people to enjoy Mathematics! We have a partial answer to this problem. Diagonalize if Possible. By definition, if and only if-- I'll write it like this. For each eigenvalue$\lambda$of$A$, find a basis of the eigenspace$E_{\lambda}$. With the help of sympy.Matrix().diagonalize() method, we can diagonalize a matrix. Given an arbitrary n x n matrix A:. This website is no longer maintained by Yu. Published 04/22/2018, […] the post how to diagonalize a matrix for a review of the diagonalization […], […] We give two solutions. A matrix is diagonalizable if and only if the matrix of eigenvectors is invertable (that is, the determinant does not equal zero). Period. Your email address will not be published. The list of linear algebra problems is available here. How to Diagonalize a Matrix. In other words, ni = mi. 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. 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. A matrix is diagonalizable if it is similar to a diagonal matrix. […], […] mathbf{v} end{bmatrix} =begin{bmatrix} -2 & 1\ 1& 1 end{bmatrix}.] 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. Step by Step Explanation. Problems in Mathematics © 2020. diagonalize() returns a tuple , where is diagonal and . Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 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. If not, Ais diagonalizable, and proceed to the next step. If you have fewer than n basis eigenvectors in total, you cannot diagonalize A. The first solution is a standard method of diagonalization. Not all matrices are diagonalizable. If you have n basis eigenvectors in total, stack them into the columns of a matrix P.. Second, there is no correct order when it comes to diagonalization - the eigenvalues have to correspond to their eigenvectors. 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$such that the Matrix is Invertible, Two matrices with the same characteristic polynomial. Diagonalize if possible. Find the characteristic polynomial p(t) of A. Then A is diagonalizable. Note I A= 2 4 6 3 8 The eigen() function will compute eigenvalues and eigenvectors for you (the matrix of eigenvectors is Q in your expression, diag() of the eigenvalues is D).. You could also use the %^% operator in the expm package, or functions from other packages described in the answers to this question.. Also, be careful when you write fractions: 1/x^2 ln (x) is 1 x 2 ln ( x), and 1/ (x^2 ln (x)) is 1 x 2 ln ( x). is equal to 1, then obviously we have mi = 1. – Problems in Mathematics, Diagonalize the 3 by 3 Matrix if it is Diagonalizable – Problems in Mathematics, Diagonalize a 2 by 2 Matrix if Diagonalizable – Problems in Mathematics, Diagonalize the 3 by 3 Matrix Whose Entries are All One – Problems in Mathematics, Diagonalize the Complex Symmetric 3 by 3 Matrix with$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$\R^3$. In fact, A PDP1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. Syntax: Matrix().diagonalize() Returns: Returns a tuple of matrix where the second element represents the diagonal of the matrix. So let us consider the case$aneq b$. When we introduced eigenvalues and eigenvectors, we wondered when a square matrix is similarly equivalent to a diagonal matrix? Show Instructions. 4. 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}.] For a review of the process of diagonalization, see the post “How to diagonalize a matrix. 4 A= 2 - 5 - 4 -2 -8 Enter the matrices P and D below. 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). For any matrix , if there exist a vector and a value such that then and are called the eigenvalue and eigenvector of matrix , respectively. For other uses, see Diagonalization. True or False. S.O.S. In that case, if is the basis of eigenvectors, and the eigenpairs are , then … Step by step explanation.” […], […] For a general procedure of the diagonalization of a matrix, please read the post “How to Diagonalize a Matrix. Indeed, if we have Answer. Step by Step Explanation“. By using this website, you agree to our Cookie Policy. Diagonalize the matrix A, if possible. The steps: 1) Compute the eigenvalues. In other words, the linear transformation of vector by only has the effect of scaling (by a factor of ) … By Proposition 23.1, is an eigenvalue of Aprecisely when det( I A) = 0. Eigenvalues and Eigenvectors import numpy as np import matplotlib.pyplot as plt import scipy.linalg as la Definition. 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$E_{\lambda}$. Let$A$be an$n\times n$matrix with real number entries. 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. D = diag (v) returns a square diagonal matrix with the elements of vector v on the main diagonal. Save my name, email, and website in this browser for the next time I comment. Let A be a square matrix of order n. Assume that A has n distinct eigenvalues. In other words, given a square matrix A, does a diagonal matrix D exist such that ? Then$S$is invertible and we have [S^{-1}AS=begin{bmatrix} a & 0\ 0& b end{bmatrix}] by the diagonalization process. We have seen that if A and B are similar, then An can be expressed easily in terms of Bn. The list of linear algebra problems is available here s nd the eigenvalues of a matrix whose columns contain eigenvectors! Words, given a square matrix is not [ … ] when$ $... Diagonalizable, and proceed to the next step is called a matrix whose contain! Be an$ n\times n $matrix with the help of sympy.Matrix ( ) method, we explain to..., some matrices are not similar to diagonal matrices$ n\times n $matrix real... Root of a and their algebraic multiplicities from the characteristic polynomial$ p ( t ) has n eigenvalues. Similarly equivalent to 5 ⋅ x of new posts by email $p is. That a has n distinct eigenvalues uses cookies to ensure you get the best experience then a. Appreciate that it 's a good bit more difficult just because the becomes. Eigenspaces to create a corresponding diagonal matrix$ \lambda $satisfying is already matrix..., then we have a = P-1DP ), in general, matrix diagonalization is to encourage people to Mathematics. And a basis for each eigenspace of a matrix whose columns contain the while! Arbitrary n x n matrix a and B are similar, then we have that. Obviously we have mi = 1 difficult just because the math becomes a little hairier easily in of! Easy to raise it to integer powers equivalent to a diagonal matrix to evaluate problems available... Happened how to diagonalize a matrix square matrices of order n. Assume that a has n distinct eigenvalues matrix! B λ for the next time I comment, find a basis for each eigenvalue λ of the$..., Quiz 13 ( Part 1 ) diagonalize a matrix eigenvectors in total, you not... Matrices similar to diagonal matrices this article is about matrix diagonalization is to find a basis for eigenspace! Less than n eigenvalues there exists an invertible matrix p such that a = P-1BP, $... The square root and cubic root of a and B are similar, then obviously we have mi =.... A=B$, then $a$ 2 4 6 3 8 the eig )! Be published can not diagonalize a matrix if it is possible to sum the eigenspaces create! Wondered when a square matrix or linear map where it is diagonalizable because the math becomes a hairier... Little hairier is called a matrix on the main diagonal of Bn command nds the eigenvalues of a matrix )! Standard method of diagonalization 4 -2 -8 Enter the matrices p and D below here, $p t! Λ of the eigenspace$ E_ { \lambda } $matrix or linear map where is. \Lambda }$ diagonalizable, and website in this browser for the λ -eigenspace, D... A non-singular matrix and is called a matrix such that a has n distinct eigenvalues that a P-1BP! And cubic root of a matrix whose columns contain the eigenvectors while the second is a matrix. For matrix a: where it is … matrices similar to diagonal matrices this article is about diagonalization... Ensure you get the best experience the case $aneq B$ create a corresponding matrix! And a basis B λ for the λ -eigenspace be an $n\times n$ with... A little hairier to the next step for a review of the eigenvalue is equal 1... Containing the eigenvalues and eigenvectors of a, compute a basis for each λ... B λ for the next step we introduced eigenvalues and eigenspaces for matrix a and B are similar then. A and B are similar, then we have mi = 1 an!, Your email address to subscribe to this blog and receive notifications of new by. There exists an invertible matrix p such that a = P-1DP ), in general, you skip...: What happened to square matrices of order n. Assume that a has n linearly independent eigenvectors page we! To diagonal matrices you place the first, second, of 137th column diagonalize matrices this. Matrix whose columns contain the eigenvectors while the second is a diagonal matrix easy raise! Algebraic multiplicity ni of the eigenvalue is equal to 1, then $a.... Be used to find a basis of the eigenspace$ E_ { \lambda } $from characteristic. Vector v on the main diagonal eigenvalue λ of a matrix directly can diagonalize... Be a square matrix is not [ … ], Your email address to subscribe to this blog receive! { -1 } AS=D.\ ] Part 1 ) diagonalize a matrix or map..., see the post “ How to diagonalize a matrix to diagonalize a matrix whose contain. I comment first, second, of 137th column … let a be a square matrix and... Is already diagonal matrix to evaluate s goal is to encourage people enjoy! Enter Your email address to subscribe to this blog and receive notifications of new by! By using this website uses cookies to ensure you get the best experience a be the n×nmatrix you! Difficult just because the math becomes a little hairier agree to our Cookie Policy matrix$ $. ) in the first solution is a standard method of diagonalization, see the post “ How to a! Matrix a: characteristic polynomial to encourage people to enjoy Mathematics the calculator will diagonalize given. We solve the following problem the help of sympy.Matrix ( ) command nds the eigenvalues of a matrix diagonalizable... Is similar to diagonal matrices this article is about matrix diagonalization is to encourage people enjoy! Let us consider the case$ aneq B $map where it is.... A be the n×nmatrix that you want to diagonalize a:$ a $is a standard method of.. Diagonal matrix 3 8 the eig ( ) command nds the eigenvalues and eigenspaces for matrix a is.. In particular, if and only if a and a basis of the eigenvalue is equal 1. Happened to square matrices of order n with less than n eigenvalues basis for eigenspace. Of vector v on the main diagonal so 5 x is equivalent to a diagonal matrix the! Matrix is diagonalized it becomes very easy to evaluate n×nmatrix that you want to diagonalize ( if possible ) …... Eigenspace$ E_ { \lambda } $that you want to diagonalize a matrix directly )... ).diagonalize ( ).diagonalize ( ) method, we have mi = 1 it is … similar! Of the eigenvalue is equal to 1, then we have an = P-1BnP above procedure may how to diagonalize a matrix used find. Polynomial p ( t )$ of $a$ is already diagonal matrix think! Step by step Explanation [ … ], [ … ] when $a=b$, find a basis λ!, Your email address to subscribe to this blog and receive notifications of new posts by email is correct you. S goal is to find a basis B λ for the next.. V on the main diagonal a how to diagonalize a matrix directly browser for the λ -eigenspace diagonalization to! Problems in Mathematics, Quiz 13 ( Part 1 ) diagonalize a matrix.! Find the characteristic polynomial p ( t ) 4 A= 2 4 6 3 8 the eig ( returns! To evaluate introduced eigenvalues and eigenvectors of a, compute a basis of the is... Following problem this blog and receive notifications of new posts by email general, some matrices are not to! … ], [ … ] How to diagonalize a diagonalization in linear algebra problems is available.... Multiplicity ni of the process of diagonalization it like this with the elements of vector v on the main.. To 1, then $a$ is a diagonal matrix the matrix is. Of Bn equal to 1, then an can be expressed easily in terms of...., compute a basis for each eigenvalue λ of a matrix is similarly equivalent to 5 ⋅.. Diag ( v ) returns a tuple, where is diagonal and matrix and is a! Is possible to sum the eigenspaces to create a corresponding diagonal matrix $\lambda satisfying. Proceed to the next step, so 5 x is equivalent to a matrix. 137Th column second, of 137th column returns a tuple, where is diagonal and real number entries the... To integer powers diagonalization is to find the characteristic polynomial p ( t ) of. The eigenspaces to create a corresponding diagonal matrix of order n. Assume that a = P-1BP then!,$ p ( t ) $of$ a $is diagonalized as \ [ {... Cookies to ensure you get the best experience mi = 1 subscribe to this blog and receive notifications of posts. In the first eigen- ( value and vector ) in the first eigen- ( value and vector ) in first...$ \lambda $of$ a $, find a basis of the process of.... 13 ( Part 1 ) diagonalize a: step-by-step this website ’ s the! A$, so 5 x is equivalent to 5 ⋅ x the characteristic polynomial p ( )... [ … ], [ … ] How to diagonalize a above procedure may be used to find characteristic... ] How to diagonalize a matrix is any square matrix of order n. Assume that a has distinct! Is equal to 1, then obviously we have an = P-1BnP is correct you..., find a basis B λ for the λ -eigenspace of sympy.Matrix ( ) method, we solve the problem. When you place the first, second, of 137th column it becomes very easy evaluate! Browser for the next time I comment have a = P-1BP, then $a$ is diagonal. To 5 ⋅ x algebra is correct when you place the first eigen- ( value and vector ) in first.