What is Canonical form: Definition and 40 Discussions

In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and which allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example:

Jordan normal form is a canonical form for matrix similarity.
The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix.In computer science, and more specifically in computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In this context, a canonical form is a representation such that every object has a unique representation (with canonicalization being the process through which a representation is put into its canonical form). Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms.
Despite this advantage, canonical forms frequently depend on arbitrary choices (like ordering the variables), which introduce difficulties for testing the equality of two objects resulting on independent computations. Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form.
Canonical form can also mean a differential form that is defined in a natural (canonical) way.

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  1. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  2. PainterGuy

    Why is it giving me different observable canonical form?

    Hi, I found the above observable canonical form using this source: https://www.mathworks.com/help/control/ug/canonical-state-space-realizations.html#mw_a76b9bac-e8fd-4d0e-8c86-e31e657471cc I'm almost certain that I did do it correctly. But the code below gives me different values for B, C...
  3. M

    Control Theory: Derivation of Controllable Canonical Form

    Hi, I was recently being taught a control theory course and was going through a 'derivation' of the controllable canonical form. I have a question about a certain step in the process. Question: Why does the coefficient ## b_0 ## in front of the ## u(t) ## mean that the output ## y(t) = b_0 y_1...
  4. A

    I How to find the canonical form of a straight line equation in space?

    Hi friends How exactly do we change the general equation of a line in space( given two intersecting planes) into the canonical form Thanks
  5. George Keeling

    I What is the canonical form of the metric?

    I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes: Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...
  6. J

    Maple Find Jordan Canonical Form with Maple

    Hi all! I have to show that the matrix 10x10 matrix below is nilpotent, determine its signature, and find its Jordan canonical form. [-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13] [15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74] [-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47] [1...
  7. Sanchayan Ghosh

    I Canonical form derivation of (L1'AL1)

    Hello everyone, I actually had a problem with understanding the part where they have defined L'AL = Λ. There, they have taken γΛγ1 = Σy2λ = 1. Why have they taken that? Is it arbitary or does it come as a result of a derivation? Thank you
  8. nightingale123

    Finding the Jordan canonical form of a matrix

    Homework Statement About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know. $$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$ Find the a) Jordan canonical form of ##A## b) characteristic polynomial c) minimal polynomial d) ##dim\,kerA## When: case 1: we know that ##A## is...
  9. Mr Davis 97

    Finding the Jordan canonical form of a matrix

    Homework Statement Find the Jordan canonical form of the matrix ## \left( \begin{array}{ccc} 1 & 1 \\ -1 & 3 \\ \end{array} \right)##. Homework EquationsThe Attempt at a Solution So my professor gave us the following procedure: 1. Find the eigenvalues for each matrix A. Your characteristic...
  10. A

    Expressing a quadratic form in canonical form using Lagrange

    Problem: Express the quadratic form: q=x1x2+x1x3+x2x3 in canonical form using Lagrange's Method/Algorithm Attempt: Not really applicable in this case due to the nature of my question The answer is as follows: Using the change of variables: x1=y1+y2 x2=y1-y2 x3=y3 Indeed you get...
  11. B

    Finding Jordan canonical form of these matrices

    Homework Statement For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A. The matrices are: ## a) \begin{pmatrix} 1 & 1\\ -1 & 3...
  12. evinda

    MHB Exploring Canonical Form of Linear Programming

    Hello! (Wave) A linear programming problem is in canonical form if it's of the following form: $$\pm \max (c_1 x_1+ \dots + c_n x_n) , c_1, \dots, c_n \in \mathbb{R} \\ Ax=b, A \in F^{m \times n}, x=\begin{bmatrix} x_1\\ \dots\\ \dots \\ x_n \end{bmatrix}, b=\begin{bmatrix} b_1\\ \dots\\...
  13. T

    Significance of Jordan Canonical Form

    I just finished a course on linear algebra which ended with Jordan Canonical Forms. There were many statements like "Jordan canonical forms are extremely useful," etc. However, we only learned a process to put things into Jordan canonical form, and that was it. What makes Jordan canonical...
  14. M

    MHB Canonical form of the hyperbolic equation

    Normal form of the hyperbolic equation Hey! :o I am looking at the following in my notes: $$a(x,y) u_{xx}+2 b(x,y) u_{xy}+c(x,y) u_{yy}=d(x,y,u,u_x,u_y)$$ $$A u_{\xi \xi}+ 2B u_{\xi \eta}+C u_{\eta \eta}=D$$ $$A=a \xi_x^2+2b \xi_y \xi_x+c \xi_y^2 \ \ \ (*)$$ $$B=a \xi_x \eta_x +b \eta_x...
  15. B

    Cylindrically symmetric line element canonical form

    Hello, What is the most general cylindrically symmetric line element in the canonical form? Best regards.
  16. C

    MHB How do you find frobenius canonical form of a matrix?

    the actual problem is to show that the given matrix is similar to companion matrix here is the companion matrix Companion matrix - Wikipedia, the free encyclopedia ---------------- i know that if same frobenius canonical form then similar but i don't even know how to find the frobenius...
  17. Sudharaka

    MHB Normal Form and Canonical Form of a Quadratic

    Hi everyone, :) Take a look at the following question. Problem: Determine which of the following quadratic functions \(q_1,\,q_2:\,V\rightarrow\Re\) is positive definite and find a basis of \(V\) where one of \(q_1,\,q_2\) has normal form and the other canonical...
  18. A

    Transition matrix and rational canonical form

    Homework Statement I want to find the transition matrix for the rational canonical form of the matrix A below. Homework Equations The Attempt at a Solution Let ##A## be the 3x3 matrix ##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}## The...
  19. R

    What Determines the Jordan Canonical Form of a Matrix?

    Homework Statement Let A be an nxn matrix (of real or complex components) and J=\left(\begin{array}{c} λ & 1 & 0 & 0\\ 0 & λ & 1 & 0\\ & & ... & \\ 0 & 0 & λ & 1\\ 0 & 0 & 0 & λ &\end{array}\right) \,with\, λ \in ℂ Show that there is S = \left(\begin{array}{c} v1 & v2 & ...
  20. B

    Proving Jordan Canonical Form for Similarity of Matrices with Same Polynomials

    I have to prove the following result: Let A,B be two n×n matrices over the field F and A,B have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than 3, then A and B are similar. I have to use the following result: If A,B are...
  21. I

    Putting a matrix into Rational Canonical Form

    I'm trying to put a matrix into RCF, and I keep running into problems. I've checked my work a few times, so I think I must be making a conceptual error. Here's what I've got: $$A=\left( \begin{matrix}2 & -2 & 14 \\ 0 & 3 & -7 \\ 0 & 0 & 2\end{matrix}\right)\quad \text{ so }\quad xI-A=\left(...
  22. D

    MHB Making Jordan Canonical Form: Box in Bottom Lambdas

    I am trying to make this into jordan canonical form. How can I box in the bottom two lambdas? $$ \left[\begin{array}{ccc} \begin{array}{cccc|} \lambda & 1 & 0 & \\ & \lambda & 1 & 0\\ & & \lambda & 1\\ & & & \lambda\\\hline \end{array} & & \\ & \begin{array}{c|} \lambda\\\hline \end{array}...
  23. C

    Markov transition matrix in canonical form?

    As I understand, a Markov chain transition matrix rewritten in its canonical form is a large matrix that can be separated into quadrants: a zero matrix, an identity matrix, a transient to absorbing matrix, and a transient to transient matrix. The zero matrix and identity matrix parts are easy...
  24. E

    Understanding Parabolas: What's the Importance of Canonical Form?

    Okay I want to be clear, this is not an homework question. But I have a hard time understanding the concept of what I'm going to show you. The teacher goes way to fast for me, honestly, and when we ask a question it's just like he doesn't want to take the time to make us understand, kinda...
  25. L

    Matrix Canonical Form: Rational and Jordan Methods

    Homework Statement Given the invariant factors of the matrix λI - A are d1 to d5 = 1, d6 = (λ-3)(λ-2)2 , d7 = (λ-3)2(λ-2)2 Display the rational canonical form of A Display the Jordan canonical form of A The Attempt at a Solution I'm half guessing here, but we have for d6 : λ3 - 7λ2 +...
  26. O

    Finding the canonical form of a quadratic form.

    could someone please explain briefly what the problem is with my method of finding such canonical forms. The method we've been taught is to find the canonical form by performing double row/column operations on the matrix representation of quadratic form until we get to a diagonal matrix, and...
  27. M

    Are A and B Similar Over a Subfield if They Are Similar Over an Extension Field?

    Hi, I'm just new here, I don't know if I'm on the right thread.:D Homework Statement Let F be a subfield of K. A, B be elements of Mn(F). Show that if A and B are similar over K, then A,B are similar over F. (Hint: what can be said about the rank of f(C(f(x)^m))^n? about the rank of...
  28. P

    Jordan Canonical Form: Equivalent Operators A and B?

    Homework Statement Are the operators specified by the matrices: A = \left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right] B = \left[\begin{array}{ccc} 4 & 1 & -1 \\ -6 & -1 & -3 \\ 2 & 1 & 1 \end{array}\right] equivalent? Homework Equations See...
  29. T

    Is the Null Space the Same for (T-λI)^k and (λI-T)^k in Linear Algebra?

    Homework Statement Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K N((T-λI)^k) = N((λI-T)^k) Homework Equations T(-v) = -T(v) N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all...
  30. S

    Minimum polynomial and canonical form

    Homework Statement Hi all. I have no clue on how to do this problem because I missed the class where he covered this so could someone please walk me through it. A = [2 2 -5; 2 7 2; -5 -15 -4] where ; means new column p(x) = (x-3)(x-1)^2 1) what are the choices of m(x) 2) find...
  31. J

    Canonical form and change of coordinates for a matrix

    Hello! I'm trying to do some linear algebra. I have an insane Russian teach whose English is, uh, lacking.. so I'd appreciate any help with these I can get here! Homework Statement Find the canonical forms for the following linear operators and the matrices for the corresponsing change of...
  32. A

    How Do You Compute the Matrix Q for Rational Canonical Form?

    Hi, can someone show me how one would go about finding the matrix Q in Q^(-1) A Q = RationalCanonicalForm(A). Please demonstrate using the example {4, 1, 2, 0} {-4, 0, 1, 5} = A {0, 0, 1, -1} {0, 0, 1, 3} where the characteristic polynomial is (x-2)^4 and the minimal polynomial is...
  33. F

    Simple jordan canonical form question

    two part question to which i have answered the first part and am stuck on the 2nd part find r and invertible real matrices Q and P such that Q-1AP=(Ir,0),(0,0) where each 0 denotes a matrix of zeroes(not necessarily the same size in each case) second part being paying special attention to...
  34. J

    Exploring Jordan Canonical Form: Clear Proof and Resources

    Does anybody know of any good websites that contain a clear proof of the existence of the Jordan Canonical Form of matrices? My professor really confused me today
  35. A

    Find th Jordan canonical form of a matrix

    I've followed and understood this small example of calculating jordan forms all the way to the last line where they say "Therefore, the jordan form is...". When they say "therefore", it's NEVER obvious :smile: Anyway, I get why the diagonal entries are -1. And that a minimal polynomial (t+1)^2...
  36. C

    Are Similar Matrices Always Similar to Their Transpose?

    1. Show that two matrices A,B ∈ Mn(C) are similar if and only if they share a Jordan canonical form. 2. Prove or disprove: A square matrix A ∈ Mn (F) is similar to its transpose AT. If the statement is false, find a condition which makes it true. (I'm pretty sure that this is true and can be...
  37. K

    Why Is Ran(A - λI) an Invariant Subspace of A?

    I'm trying to teach myself math for physics (a middle aged physicist wannabee). Wikipedia's proof for the exisitence of a JC form for matrix A in Cn,n states: "The range of A − λ I, denoted by , is an invariant subspace of A" I'm having trouble seeing why any element of Ran(A − λ I) is in...
  38. P

    Canonical Form of Matrices: Understanding and Transforming

    Homework Statement Matrix: \left| \begin{array}{ccc} \-1 & -2 & 5 \\ 6 & 3 & -4 \\ -3 & 3 & -11 \end{array} \right|\] Homework Equations The Attempt at a Solution How will this matrix transferred into canonical form? What is actually canonical form?
  39. R

    Transforming a Second-Order PDE into Canonical Form: Tips and Techniques

    How do I transform a second-order PDE with constant coefficients into the canonical form? I tried to solve this problem: u_xx + 13u_yy + 14u_zz - 6u_xy + 6u_yz + 2u_xz -u_x +2u_y = 0 I wrote the bilinear form of the second order derivatives and diagonalized it. I found out that it is a...
  40. P

    What is Rational Canonical Form and How is it Used in Rings and Fields?

    Or something like that... I need definition,, explanation and examples. I have an exam in Rings and Fields on Sunday, and he used that term during the course- I have no idea what it is. I'd appreciate any help. Thanks in advance!