What is Diagonal matrix: Definition and 55 Discussions

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is







{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}
, while an example of a 3×3 diagonal matrix is












{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&7&0\\0&0&4\end{smallmatrix}}\right]}
. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.
A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.

View More On Wikipedia.org
  1. G

    I Diagonal Matrix of Stress-Energy Tensor: Why?

    I came across a statement in《A First Course in General Relativity》:“The only matrix diagonal in all frames is a multiple of the identity:all its diagonal terms are equal.”Why?I don’t remember this conclusion in linear algebra.The preceding part of this sentence is:Viscosity is a force parallel...
  2. H

    Find a matrix ##C## such that ##C^{-1} A C## is a diagonal matrix

    I’m really unable to solve those questions which ask to find a nonsingular ##C## such that $$ C^{-1} A C$$ is a digonal matrix. Some people solve it by finding the eigenvalues and then using it to form a diagonal matrix and setting it equal to $$C^{-1} A C$$. Can you please tell me from scratch...
  3. H

    Find the basis so that the matrix will be diagonal

    First of all, it is clear that we can find such a bases (the theorem is given in almost all of the books, but if you want to share some insight I shall be highly grateful.) We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as...
  4. S

    Diagonalizing a matrix given the eigenvalues

    The following matrix is given. Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1. The answer sheet reads that the diagonal matrix D is as follows: I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
  5. N

    I Block Diagonal Matrix and Similarity Transformation

    I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
  6. V

    How to find the diagonal matrix and it's dominant eigenvalue

    Homework Statement Consider the following vectors, which you can copy and paste directly into Matlab. x = [2 2 4 6 1 5 5 2 6 2 2]; y = [3 3 3 6 3 6 3 2 3 2]; Use the vectors x and y to create the following matrix. 2 3 0 0 0 0 0 0 0 0 0 3 2 3 0 0 0 0 0 0 0 0 0 3 4 3 0 0 0 0 0 0 0 0 0 3 6 6 0...
  7. Wrichik Basu

    B Why does a matrix diagonalise in this case?

    Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
  8. Z

    Finding a matrix W such that W^t*AW = D (D is diagonal matrix)

    Homework Statement A = 000 010 101 Find Eigenvalues, its corresponding eigenvectors, and find a matrix W such that W^t*AW = D, where D is a diagnol matrix.(note that W^t represents the transpose of W) Homework Equations Eigenvalues, Eigenvectors, diagnolization[/B]The Attempt at a...
  9. H

    Operator r is a diagonal matrix in position representation

    What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that? Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)
  10. Linder88

    Determine Diagonalizability of LTI System A

    Homework Statement Consider the LTI (A,B,C,D) system $$ \dot{x}= \begin{pmatrix} 0.5&0&0&0\\ 0&-2&0&0\\ 1&0&0.5&0\\ 0&0&0&-1 \end{pmatrix} x+ \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix} u $$ $$ y= \begin{pmatrix} 0&1&0&1 \end{pmatrix} x $$ Determine if A is diagonalizable Homework EquationsThe...
  11. kostoglotov

    How can e^{Diag Matrix} not be an infinite series?

    So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
  12. kostoglotov

    Help: All subspaces of 2x2 diagonal matrices

    The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices. I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix. The answer given is here, relevant answer is (b): Imgur link: http://i.imgur.com/DKwt8cN.png...
  13. C

    Arrangement of eigenvalues in a Diagonal matrix

    Homework Statement Is it necessary to arrange the eigenvalues in increasing value order? As shown in the image attached, if I arrange my eigenvalues -2, -1, 1 diagonally, my D would be 2^8 , 1, 1 diagonally. However if i arrange it as, say, -1, 1, -2, my D would be different...
  14. Seydlitz

    Proving the special property of diagonal matrix?

    Is it possible to prove the fact that any function of diagonal matrix is just a function of its element? I don't know how I could express the proof. I can prove that a multiplication of diagonal matrix will just be the multiplication of its element using summation notation, or diagonal matrix...
  15. C

    If D is a diagonal matrix, when is B D B^T diagonal?

    Hey! So here's the question: Homework Statement Let \mathbf{B} \in \mathbb{R}^{n \times n} be some square matrix we can choose and \mathbf{D} \in \mathbb{R}^{n \times n} be some given diagonal matrix with positive diagonal elements. For what matrices \mathbf{B} is the product...
  16. Fernando Revilla

    MHB Help with Diagonal Matrix for T: R2 → R2

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  17. G

    Is it possible to have a diagonal matrix with all eigenvalues = zero ?

    Homework Statement If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors? Homework Equations The Attempt at a Solution
  18. B

    Small oscillations: diagonal matrix

    Homework Statement I'm solving an exercise about small oscillations. I name T the kinetic matrix and $H$ the hessian matrix of potential. The matrix \omega^2 T- H is diagonal and so find the auto-frequencies is easy! But I have a problem with normal modes. The lagrangian coordinates are two...
  19. matqkks

    MHB Why do we need to convert to a diagonal matrix?

    Apart from simplifying matrix powers, why do we want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices. Thanks for any answers.
  20. matqkks

    Why do we need to convert to a diagonal matrix?

    Apart from simplifying matrix powers, why do we want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices. Thanks for any answers.
  21. J

    Solveing differential equations system using diagonal matrix

    Homework Statement Solve this system of differential equations \begin{equation} x'_1=5x_1 + 2 x_2 - x_3 \\ x'_2=-2x_1 + x_2 - 2x_3 \\ x'_3=-6x_1 - 6 x_2 \end{equation} Homework Equations The Attempt at a Solution This is my first time solving a problem like this and I just...
  22. N

    Null space and eigenspace of diagonal matrix

    Homework Statement I am working on a problem where I made a matrix representation of a linear transformation and I am asked what is the eigenspace for a particular eigenvalue. Homework Equations The Attempt at a Solution The problem for me is, I came out with a diagonal...
  23. C

    Find Basis for diagonal matrix

    I'm not sure how to start this problem. All i know is a diagonal matrix consists of all 0 elements except along the main diagonal. But how do I even find a basis for this?
  24. S

    Spectral decomposition of a diagonal matrix

    Homework Statement I have J=\begin{bmatrix} \frac{\pi}{2}&0&0\\ 1&\frac{\pi}{2}&0\\ 0&1&\frac{\pi}{2}\\ \end{bmatrix} I need to find \sin(J) \text{ and } \cos(J) \text{ and show that } \sin^{2}(J)+\cos^{2}(J)=I Homework Equations The Attempt at a Solution I have the...
  25. C

    Find linear transformation using diagonal matrix

    Homework Statement Define L:R3-->R3 by L(x,y,z)=(y-z,x+z,-x+y). A. Show that L is self-adjoint using the standard orthonormal basis B of R3. B. Diagonalize L and find and orthogonal basis B of R3 of eigenvectors of L and the diagonal matrix. C. Considering only the eigenvalues of L...
  26. S

    Combine upper, lower, and diagonal matrix?

    If you have a matrix that is a combination of a diagonal, upper, and lower matrix - what is the best way to solve it? (I use MATLAB for matrix work) Example: 5 1 0 0 x1 1 -1 5 1 0 x2 2 0 -1 5 1 x3 3 0 0 -1 5 x4 4 Is it possible to solve...
  27. W

    What can commute with a diagonal matrix?

    I have two matrices which commute, one of which is definitely diagonal: \textbf{B}diag\{\underline{\lambda}\} = diag\{\underline{\lambda}\}\textbf{B} and I want to know what I can say about \textbf{B} and/or \underline{\lambda}. Specifically, I feel that either one or both of the following...
  28. P

    Finding the Inverse of a Diagonal Matrix in Terms of Eigenvalues

    Diagonalizing an N × N matrix H involves writing it as H = UDU† where D is a diagonal matrix, with diagonal elements equal to the eigenvalues of the matrix H, and U is a unitary matrix. We may write: D= (λ1 0 0 ... 0) (0 λ2 0 ... 0) (0 0 λ3... 0) (... ... ... ... λn)...
  29. fluidistic

    Linear algebra, basis, diagonal matrix

    Homework Statement Write the A matrix and the x vector into a basis in which A is diagonal. A=\begin{pmatrix} 0&-i&0&0&0 \\ i&0&0&0&0 \\ 0&0&3&0&0 \\ 0&0&0&1&-i \\ 0&0&0&i&-1 \end{pmatrix}. x=\begin{pmatrix} 1 \\ a \\ i \\ b \\ -1 \end{pmatrix}. Homework Equations A=P^(-1)A'P. The...
  30. M

    Solving Square Matrix Similarity to Diagonal Matrix

    Homework Statement A square matrix A (of some size n x n) satisfies the condition A^2 - 8A + 15I = 0. (a) Show that this matrix is similar to a diagonal matrix. (b) Show that for every positive integer k >= 8 there exists a matrix A satisfying the above condition with tr(A) = k. Homework...
  31. M

    How can a column vector be transformed into a diagonal matrix?

    I think this is a pretty simple question. I need a transformation that will take a Column vector e.g.: <a,b,c> and turn it into a 3x3 matrix where a is in position 1,1 and b in position 2,2 and c in position 3,3. i.e.: a diagonal matrix. Any help?
  32. R

    Effect on eigenvalues of multiplying by a diagonal matrix

    Hi, While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix D^{H}A^{H}AD where the matrix A is known and the matrix D is a diagonal matrix whose entries contain the variables that need to be optimized (those...
  33. T

    Diagonalization of an almost diagonal matrix

    Homework Statement If we have a n x n matrix with 1 on the diagonal entries apart from the ith column which has a -1. As well as this ith row can have any real number in each entry. Other than this the matrix is 0 everywhere. Show this matrix is diagonalisable. Homework Equations...
  34. P

    Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

    Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...
  35. C

    Nearest block diagonal matrix to a given matrix

    Suppose I have a matrix that I want to reduce to block diagonal form. Obviously, the block diagonal form is not unique as each of the diagonal blocks is defined only to within a unitary rotation. So I want to find the block diagonal matrix that is closest to the original matrix in terms of the...
  36. J

    Center of the general linear group is diagonal matrix proof

    Homework Statement center of the general linear group is diagonial matrix proof Homework Equations The Attempt at a Solution i write out a n by n matrix and multiply left by a and right by a^-1 and show that it is the same. I think it can force the matrix to be diagonal but i...
  37. J

    Comp Sci C++ determinant of diagonal matrix

    Homework Statement well, my assignment was to make a gauss elimination, so now i need to compute the determinant of an n by n diagonal matrix variable rows = number of equations variable i = random integer matrix A[100][100] dummy matrix A2[100][100] Homework Equations det[A]...
  38. P

    Find orthogonal P and diagonal matrix D

    Homework Statement A= [1 -1 0] [-1 2 -1] [0 -1 1] find orthogonal matrix P and diagonal matrix D such that P' A P = D Homework Equations The Attempt at a Solution i got eigenvalues are 0, 1, 3 which make D=[0 0 0; 0 1 0; 0 0 3] how to find P. because in...
  39. T

    Using Eigenvectors to produce a Diagonal matrix

    Homework Statement If A=[{5,3},{-2,-2}], find the eigenvectors of A. Using these eigenvectors as matrix P, find P-1 and thus prove P-1AP is diagonal. Homework Equations None The Attempt at a Solution So i can get the eigenvectors to be <3,-1> and <1,-2> corresponding to eigenvalues 4...
  40. R

    Linear Algebra- Diagonal Matrix

    Homework Statement Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1 a) Show that D is idempotent b) Show that if X is a nonsingular matrix and A=XD(X)-1 , then A is idempotentHomework Equations The Attempt at a Solution a) I tried it, and it works for a specific...
  41. Z

    Representation by a diagonal matrix question

    Homework Statement Let T be the linear operator on R3 that has the given matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. a) Determine whether T can be represented by a diagonal matrix, and b) whenever possible, find a diagonal matrix and a basis of R3 such that T is represented...
  42. J

    Singular values of a matrix times a diagonal matrix

    Hi, I have been struggling with this problem for a while, and I have not found the answer in textbooks or google. Any help would be very much appreciated. Suppose I know the singular value decomposition of matrix B, which is a singular, circulant matrix. That is, I know u_i, v_i, and...
  43. S

    How to find invertible matrix and diagonal matrix

    Homework Statement Find an invertible matrix P and a diagonal matrix D such that D=P^(-1)AP. A= (−1 2 −2) (−2 3 −2) ( 2 −2 3) Homework Equations The Attempt at a Solution the eigenvalues are 1, 1, and 3 the eigenvector I've found so far is for the eigenvalue 3...
  44. J

    Proving Eigenspace Correspondence for Similar Matrices

    Hi I have this question for my Linear Algebra class and I can't seem to figure it out. Let A and B be n x n matrices such that B = (P^-1)AP and let lambda ne an eigenvalue of A (and hence of B). Prove the following results: (a) A vector b in R^n is in the eigenspace of A corresponding to...
  45. J

    Isomorphic diagonal matrix spaces

    Homework Statement Is The space P2 is isomorphic to the space of all 3 × 3 diagonal matrices. Homework Equations The Attempt at a Solution I know that P2 is isomorphic to vectors with 3 components so i think this statement is true, is it?
  46. A

    When is the Jacobian of a function a diagonal matrix

    Homework Statement Let f(x,y,z)=(exp(x),cos(y),sin(z)).Compute the Jacobian J(f) of f . In general ,when will the Jacobian J(g) of a function g(x,y,z) be a diagonal matrix ? Homework Equations The Attempt at a Solution I am not quiet sure about this question for J(f) i found...
  47. B

    Physical reason for diagonal matrix

    Whats the physical reason for a diagonalized matrix with the eigenvalues of the system as the values? Reading from wiki, it seems that its something to do with the Schrodinger Eqn, but I don't follow that. If someone could explain the point of it to me in plain English (or as close as it...
  48. C

    Find a diagonal matrix D such that the tridiagonal matrix T

    Homework Statement Homework Equations For a symmetric matrix B=B' where ' is the transpose. The Attempt at a Solution Since we know that for a symmetric matrix, B = B' I attempted to substitude that in and tried to solve for D. DTD-1 = (D-1)'T'D DT = D-1T'DD D =...
  49. R

    MATLAB Create Matrix without For Loop in MatLab

    I am trying to create the following Matrix without the use of a for loop. 1 2 1 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 Etc. Is there a simple way of doing this Edit: Using Matlab
  50. H

    Matrix Identify involving Diagonal Matrix

    If V is diagonal, it is easy to show: (V + V^{-1})^{-1} = V(V^2 + I)^{-1} by multiplying both sides by: (V + V^{-1}) But, I'm wondering if there is a way to derive the RHS from the LHS. Since diagonal matrices behave like scalars, I used a scalar analogy: (x + 1/x)^{-1} = ((x^2...