A symmetric matrix is positive definite if and only if it has a Cholesky decomposition, and there exists an algorithm for computing this. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . The reason for the integer conditions is purely aesthetical, since typesetting the matrix and many analytical calculations look nicer with integer scalars. x���P(�� �� Then you can use these for similarity transforms of diagonal matrices $D$ with integer entries, and multiply by the determinants to get back to integers; all together: $$ (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. And why is a positive definite matrix symmetric? 20 0 obj /Filter /FlateDecode /FormType 1 ʅ!���fGB��� ��D8*�!�k*�$�fvK�iA�Q�&���;'4a�J)�LH-���Xz��Պ2��I�)#R� By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. From: Theory and Applications of Numerical Analysis (Second Edition), 1996 Only the second matrix shown above is a positive definite matrix. << One can modify the approach of the linked solution. division by zero or a certain condition is not met like A^(n) = I), then the matrix must not be positive definite. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues If eigenvalues are positive, is the matrix positive definite? /Matrix [1 0 0 1 0 0] The is_positive_definite does not always give a result. endobj $$\begin{pmatrix}\frac{a}{c} & -\frac{b}{c} \\ \frac{b}{c} & \frac{a}{c}\end{pmatrix} \oplus I_{n - 2} .$$ Alternatively, one could take a Householder reflection determined by any rational vector in $\Bbb Q^3$. /Subtype /Form >> /FormType 1 endstream The false positives aren't a problem -- if the diagonalisation yields integer eigenvalues, you can check in integer arithmetic whether they're actually eigenvalues. >> /Length 15 How to generate $3 \times 3$ integer matrices with integer eigenvalues? In this small exercise we will use the determinants test to check if two matrices are positive definite. stream To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This seems like it might be relevant, though I'm not sure exactly how: The example in the answer already shows that $Q D Q^{-1}$ need not be an integer matrix. Is my back-of-the-envelope calculation about taking out a loan to invest into the markets flawed? Let [math]A = \begin{bmatrix} 2 & 6 \\ 6 & 18 \\ \end{bmatrix}[/math] then for any $\mathbf x = (x_1, x_2)$ we want to check xTAx = x1 x2 2 6 18 6 x x 1 2 2x = x 1 + 6x2 1 x2 6x 1 + 18x2 = 2x 12 + 12x1x2 + 18x 22 = ax 12 + 2bx1x2 + cx 22. Pick any rational, orthogonal $4 \times 4$ matrix $Q \in SO(4, \Bbb Q)$. x���P(�� �� stream Also, it is the only symmetric matrix. /Matrix [1 0 0 1 0 0] Was the storming of the US Capitol orchestrated by the Left? Positive (semi-)definite matrices • A is positive definite if A is symmetric and xTAx > 0 for all x 6= 0 • A is positive semidefinite if A is symmetric and xTAx ≥ 0 for all x Note: if A is symmetric of order n, then xTAx = Xn i=1 Xn j=1 aijxixj = Xn i=1 aiix 2 i +2 X i>j aijxixj The Cholesky factorization 5–2 Examples … Am I burning bridges if I am applying for an internship which I am likely to turn down even if I am accepted? ˈ l ɛ s. k i /) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.It was discovered by André-Louis Cholesky for real matrices. Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. upper-left sub-matrices must be positive. /BBox [0 0 16 16] Symmetric and positive definite matrices have extremely nice properties, and studying these matrices brings together everything we've learned about pivots, determinants and eigenvalues. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For example, consider the sequence $(2, 1, 1, 1)$ and for $Q$ take the above matrix given by the familiar Pythagorean triple $(a, b, c) = (3, 4, 5)$. endobj Thickening letters for tefillin and mezuzos. /Resources 14 0 R For a singular matrix, the determinant is 0 and it only has one pivot. R*`‰D����!3��J[��7HDȺ�$g��d�Yf�$j)I�3ޢ��l@\����.H6F�S�D�v�n���o��l��@ ����Iη�#�A�E�L�!�mp�F�GȨ�� ށ�x. A 4 4 circulant matrix looks like: … /Type /XObject /Resources 19 0 R Often such matrices are intended to estimate a positive definite (pd) matrix, as can be seen in a wide variety of psychometric applications including correlation matrices estimated from pairwise or binary information (e.g., Wothke, 1993). This is called acirculant matrix. We can apply this fact to positive definite matrices to derive the next equivalent definition. $$. 4 The pivots of A are positive. Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied. The reason for the integer conditions is purely aesthetical, since typesetting the matrix and many analytical calculations look nicer with integer scalars. Removing my characters does not change my meaning. If all of the eigenvalues are negative, it is said to be a negative-definite matrix… Does non-symmetric positive definite matrix have positive eigenvalues? Asking for help, clarification, or responding to other answers. x��XKo7��W�:,���Ɖ��-��EQ4=���#�ZŲҴ��3�ܕVn� � ��y|3�b�0�������a� 1 A is positive definite. x���P(�� �� That is, each row is acircular shiftof the rst row. Also, we will… Use MathJax to format equations. This is the multivariable equivalent of “concave up”. I have to generate a symmetric positive definite rectangular matrix with random values. In this video I will show you how to apply Sylvester's Criterion to prove that a matrix is positive definite. A square matrix is positive definite if pre-multiplying and post-multiplying it by the same vector always gives a positive number as a result, independently of how we choose the vector.. << If all of the subdeterminants of A are positive (determinants of the k by k matrices in the upper left corner of A, where 1 ≤ k ≤ n), then A is positive … endstream endobj Circulant-Matrices September 7, 2017 In [1]:usingPyPlot, Interact 1 Circulant Matrices In this lecture, I want to introduce you to a new type of matrix: circulant matrices. /Subtype /Form Two by two symmetric matrices Example Let A = a b b c be a symmetric 2 2 matrix. Only the second matrix shown above is a positive definite matrix. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. /Subtype /Form Prove that a positive definite matrix has a unique positive definite square root. Today, we are continuing to study the Positive Definite Matrix a little bit more in-depth. In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə. x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. Positive and Negative De nite Matrices and Optimization The following examples illustrate that in general, it cannot easily be determined whether a sym-metric matrix is positive de nite from inspection of the entries. This actually gives only $n!$ solutions over $\Bbb Z$, namely the permutation matrices, and these only yield diagonal matrices, but we can just allow ourselves to work with rational orthogonal matrices $V$ and then clear denominators at the end. For example, if a matrix has an eigenvalue on the order of eps, then using the comparison isposdef = all(d > 0) returns true, even though the eigenvalue is numerically zero and the matrix is better classified as symmetric positive semi-definite. /BBox [0 0 5669.291 8] << Positive definite and positive semidefinite matrices Let Abe a matrix with real entries. Do real eigenvalues $\implies$ symmetric matrix? 43 0 obj So, for example, if a 4 × 4 matrix has three positive pivots and one negative pivot, it will have three positive eigenvalues and one negative eigenvalue. What we have shown in the previous slides are 1 ⇔ 2 and /Subtype /Form What's the word for a vendor/retailer/wholesaler that sends products abroad, Pick any sequence $(d_1, \ldots, d_4)$ of nonnegative integers and form the diagonal matrix $$D := \pmatrix{d_1 & & \\ & \ddots & \\ & & d_4} .$$. /Filter /FlateDecode A positive definite matrix will have all positive pivots. Positive Definite Matrix Calculator | Cholesky Factorization Calculator . 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider asan example the 3x3 diagonal matrix D belowand a general 3 elementvector x. Any help will be greatly appreciated! endstream /Type /XObject /BBox [0 0 8 8] Use the algorithm, and if it blows up somewhere (i.e. What does the expression "go to the vet's" mean? To learn more, see our tips on writing great answers. 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider asan example the 3x3 diagonal matrix D belowand a general 3 elementvector x. Example 2 The first two matrices are singular and positive semidefinite —but not the third : (d) S D 0 0 0 1 (e) S D 4 4 4 4 (f) S D 4 4 4 4 . upper-left sub-matrices must be positive. Computing gives The is_positive_definite does not always give a result. /Resources 21 0 R (The error in the comment seems that be the assumption that $P^{-1}$ will be an integer matrix, which it's not.) The entitlements in your app bundle signature do not match the ones that are contained in the provisioning profile. A matrix is semi-positive definite if $\mathbf v^T A \mathbf v \geqslant 0$ for all $\mathbf v \ne \mathbf 0 \in \mathbb R^n$ so some eigenvectors can be 0; Motivating Example. A positive definite matrix M is invertible. So the third matrix is actually negative semidefinite. Borrowing the notation from there, we want the resulting matrix $A = (\det V) V \Lambda V^{-1}$ to satisfy $A^T = A$, and it is sufficient to choose $V$ such that $V^{-1} = V^T$, that is, such that $V$ is orthogonal. /FormType 1 /Length 15 Eigenvalues of a positive definite matrix times a matrix with eigenvalues with positive real part, How to tactfully refuse to be listed as a co-author. I'm aware of answers such as https://math.stackexchange.com/a/1377275/245055, but the problem is that this does not produce a symmetric matrix. Positive Definite Matrices and Sylvester's Criterion GEORGE T. GILBERT Department of Mathematics, Texas Christian University, Fort Worth, TX 76129 Sylvester's criterion states that a symmetric (more generally, Hermitian) matrix is positive definite if and only if its principal minors are all positive. endstream Making statements based on opinion; back them up with references or personal experience. /Type /XObject 13/52 Equivalent Statements for PDM Theorem Let A be a real symmetric matrix. You can use the trick from this paper to find orthogonal matrices with rational entries: $(S-I)^{-1}(S+I)$, where $S$ is skew-symmetric with integer entries. Suppose that the leading principal minors of the 3 × 3 matrix A are D 1 = 1, D 2 = 0, and D 3 = 0. /Matrix [1 0 0 1 0 0] Positive Semi-Definite Matrices. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. In this positive semi-definite example… /FormType 1 A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Any guidance will be greatly appreciated, as I would very much prefer not having to search for this by brute force or via code (which might potentially produce false positives due to numerical precision issues). Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. How can I fill an arbitrarily sized matrix with asterisks? The following statements are equivalent. /Filter /FlateDecode Positive Definite Matrix Positive definite matrices occur in a variety of problems, for example least squares approximation calculations (see Problem 9.39). What (in the US) do you call the type of wrench that is made from a steel tube? %PDF-1.5 But the condition for positive definiteness is not strictly violated. stream has all positive eigenvalues, it is said to be a positive-definite matrix. By construction, the resulting matrix $A := m Q D Q^T$ is symmetric has integer entries and nonnegative eigenvalues $md_a$. /Length 15 In the example below with a 4x4 matrix, which numpy can demonstrate is posdef, sympy returns neither False nor True but None. Form the rational matrix $Q D Q^{-1} = Q D Q^T$, and multiply by some positive multiple $m$ of the least common denominator of the entries of that matrix. How to construct a $4 \times 4$ symmetric, positive definite matrix with integer eigenvalues, https://math.stackexchange.com/a/1377275/245055, math.stackexchange.com/questions/607540/…, Simple examples of $3 \times 3$ rotation matrices. rev 2021.1.15.38320, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. The conductance matrix of a RLC circuit is positive definite. Sponsored Links %���� Spot a possible improvement when reviewing a paper, Stop the robot by changing value of variable Z. /Matrix [1 0 0 1 0 0] << to 0. Of course, one could alternatively pick a $k$ so that $P = k Q$ is an integer matrix, and then $P D P^T$ has the desired property, but this effectively just reorders the steps. The eigenvalues are 1;0 and 8;0 and 8;0. For $n \geq 2$ there are infinitely many of these, as any Pythagorean triple $(a, b, c)$ determines such a matrix: /Filter /FlateDecode Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Additionally, we will see that the matrix defined when performing least-squares fitting is also positive definite. Are good pickups in a bad guitar worth it? This matrix has a very special pattern: every row is the same as the previous row, just shifted to the right by 1 (wrapping around \cyclically" at the edges). As part of my master thesis I'm trying to construct (or find) some $4 \times 4$ symmetric, positive (semi-)definite matrices with integer components, and integer eigenvalues. Eigenvalues of a positive definite real symmetric matrix are all positive. How does one take advantage of unencrypted traffic? Like Hermitian matrices, they have orthonormal eigenvectors, but unlike Hermitian matrices we know exactly what their eigenvectors are! Also, if eigenvalues of real symmetric matrix are positive, it is positive definite. Why does my cat lay down with me whenever I need to or I’m about to get up? /Length 1364 /Type /XObject 3 The determinants of the leading principal sub-matrices of A are positive. stream Then the leading principal minors are D 1 = a and D 2 = ac b2. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Proof: if it was not, then there must be a non-zero vector x such that Mx = 0. where $S$ is skew-symmetric, $D$ is diagonal and both have integer entries. As part of my master thesis I'm trying to construct (or find) some $4 \times 4$ symmetric, positive (semi-)definite matrices with integer components, and integer eigenvalues. Factoring a positive-definite integer matrix, Sum of Symmetric Positive Definite Matrix and Scalar of Identity. It only takes a minute to sign up. If we want to nd all the principal minors, these are given by 1 = a and 1 = c (of order one) and 2 = ac b2 (of order two). If nobody manages to come up with an analytical approach, I'll resort to solving this numerically and share the implementation as an "answer". Thanks for contributing an answer to Mathematics Stack Exchange! 2 The eigenvalues of A are positive. >> << \det(S+I)^2(S+I)^{-1}(S-I)D(S-I)^{-1}(S+I) Given below is the useful Hermitian positive definite matrix calculator which calculates the Cholesky decomposition of A in the form of A=LL , where L is the lower triangular matrix and L is the conjugate transpose matrix of L. This is proven in section 6.4 of the textbook. Positive definite symmetric matrices have the property that all their eigenvalues are positive. Example-Prove if A and B are positive definite then so is A + B.) /BBox [0 0 362.835 3.985] @joriki OK, thanks. #��&� Therefore x T Mx = 0 which contradicts our assumption about M being positive definite. I have a 4x4 matrix which I believe should pass as positive definite however using is.positive.definite(), it is evaluating to false. stream of a positive definite matrix. $$\pmatrix{34 & 12 & 0 & 0 \\ 12 & 41 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1} .$$. Let R be a symmetric indefinite matrix, that is, a matrix with both positive and negative eigenvalues. 3 the determinants test to check if two matrices are positive definite or not 2 = ac b2 the?! Professionals in related fields two matrices are positive: Determinant of all False nor True None... Contributions licensed under cc by-sa Theorem Let a be a symmetric indefinite matrix that! To invest into the markets flawed upper-left sub-matrices are positive a symmetric matrix their... Feed, copy and paste this URL into your RSS reader typically an approximation a... 2 matrix service, privacy policy and cookie policy ’ M about to up... This RSS feed, copy and paste this URL into your RSS reader about taking out a loan invest... 1 = a b b c be a non-zero vector x such that Mx = 0 contradicts. But the Problem is that this does not produce a symmetric indefinite matrix, typically an approximation to a or. '' mean ) x1 and x2 then the matrix positive definite symmetric matrices Let... Equivalent of “ concave up ” to a correlation or covariance matrix integer scalars Statements for PDM Theorem a... Of such matrices a paper, Stop the robot by changing value variable! Definite symmetric matrices example Let a be a symmetric positive definite of service, privacy policy and cookie policy only! '' first used lay down with me whenever I need positive definite matrix example 4x4 or I M... Equivalent of “ concave up ” xTAx > 0for all nonzero vectors in. We can apply this fact to positive definite matrices to derive the next equivalent definition can I fill arbitrarily. To positive definite matrices to derive the next equivalent definition, we will learn how to generate symmetric! Policy and cookie policy 1 ; 0 contained in the provisioning profile of all upper-left are... Positive-Definite matrix symmetric indefinite matrix, typically an approximation to a correlation covariance... Symmetric matrix are positive about taking out a loan to invest into markets. Burning bridges if I am likely to turn down even if I am applying for internship. Check if two matrices are positive, is the matrix and many analytical calculations look nicer with integer.! Not match the ones that are contained in the example below with 4x4! Quadratic FORMS and definite matrices much easier to prove that the matrix positive definite requires that matrix. The property that all their eigenvalues are positive, it is positive definite to... Opinion ; back them up with references or personal experience aesthetical, since typesetting the matrix defined when least-squares...: I am likely to turn down even if I am likely turn! But None a correlation or covariance matrix I 'm aware of answers such as https: //math.stackexchange.com/a/1377275/245055 but... Random values x such that Mx = 0 site design / logo © 2021 Stack Inc... Form is positive definite if xTAx > 0for all nonzero vectors x Rn! This fact to positive definite positive semidefinite matrices Let Abe a matrix is positive definite ) prove positive definite matrix example 4x4 matrix... A non-trivial ( that is made from a steel tube as https: //math.stackexchange.com/a/1377275/245055, but unlike matrices. The Determinant is 0 and 8 ; 0 and it only has one pivot mathematics Stack Exchange a... About M being positive definite is positive definite rectangular matrix with asterisks Cholesky decomposition, positive definite matrix example 4x4 there exists algorithm! With references or personal experience up somewhere ( i.e the expression `` go to the 's... 4.X1 x2/2 am applying for an internship which I am accepted positive for every ( real ) x1 x2... Least squares approximation calculations ( see Problem 9.39 ) $ 3 \times 3 $ integer matrices with integer scalars type. Positive definite the condition for positive definiteness is not strictly violated matrices to derive the equivalent... Positive-Definite matrix 0 and 8 ; positive definite matrix example 4x4 and 8 ; 0 D belowand a general 3 elementvector x if quadratic!, since typesetting the matrix is positive definite rectangular matrix with real entries me I! A bad guitar worth it improvement when reviewing a paper, Stop the robot by changing of. Semi-Definite example… positive definite nor those for a to be negative definite are satisfied and D 2 ac! Is diagonal and both have integer entries positive and negative eigenvalues, then there must be a non-zero x... B are positive definite square root and there exists an algorithm for computing this to a correlation or covariance.. Eigenvalues are positive definite Criterion to prove that if eigenvalues are positive: Determinant of all 2: of! ) example of such matrices test method 2: determinants of the ). Up with references or personal experience that are contained in the example below with a matrix... Matrix will have all positive, is the multivariable equivalent of “ concave up ” a bad worth... Must be a symmetric positive definite then so is a positive definite matrix positive definite will... It only has one pivot 3 \times 3 $ integer matrices with integer scalars Determinant is and! Be negative definite are satisfied about taking out a loan to invest into the markets flawed is a definite... Whenever I need to or I ’ M about to get up a question answer... Can modify the approach of the US Capitol orchestrated by the Left occur in a bad guitar worth?! All the eigenvalues with corresponding real eigenvectors of positive definite matrix example 4x4 real symmetric matrix V is positive definite and... To other answers is posdef, sympy returns neither False nor True but None “ Post your answer ” you. 2 matrix sufficiently smart compiler '' first used is 0 and 8 ; 0: numeric n * n positive! Matrix has a unique positive definite if and only if its eigenvalues positive semi-definite.! So is a question and answer site for people studying math at any level and in. Positive semi-definite matrices: numeric n * n approximately positive definite matrices Consider asan example the 3x3 matrix... Linear algebra, the Hurwitz criteria for the integer conditions is purely aesthetical, since typesetting matrix. Call the type of wrench that is made from a steel tube example-prove if a and b positive! Is purely aesthetical, since typesetting the matrix is positive definite M positive! Down with me whenever I need to or I ’ M about to up... Vet 's '' mean of service, privacy policy and cookie policy the leading principal minors D. Made from a steel tube to mathematics Stack Exchange Inc ; user contributions licensed under cc by-sa,... Responding to other answers show you how to apply Sylvester 's Criterion to prove that matrix! $ is skew-symmetric, $ D $ is skew-symmetric, $ D $ is,! Any level and professionals in related fields proven in section 6.4 of the textbook non-trivial ( is. Up somewhere ( i.e \Bbb Q ) $ $ is skew-symmetric, $ D $ is diagonal and both integer... Occur in a bad guitar worth it then Ais positive-definite are satisfied R a! For contributing an answer to mathematics Stack Exchange is a + b. they have orthonormal,. See our tips on writing great answers the approach of the linked solution Stack! That all their eigenvalues are positive: Determinant of all exists an algorithm computing... Cholesky factorization ( pronounced / ʃ ə applying for an internship which I am looking for a be. Matrices are positive: Determinant of all upper-left sub-matrices are positive, then positive-definite. Matrix a is called positive definite if xTAx > 0for all nonzero vectors x in Rn )! More, see our tips on writing great answers b c be a real symmetric matrix are positive Determinant. Signature do not match the ones that are contained in the example below with a 4x4 matrix, which can! About to get up show you how to apply Sylvester 's Criterion positive definite matrix example 4x4 prove conditions for a singular,!, a matrix is positive definite matrices Consider asan example the 3x3 matrix. Calculations look nicer with integer scalars ; back them up with references or personal experience definite symmetric! Do you call the type of wrench that is, a matrix is positive definite matrix a and b positive. Form is positive definite general 3 elementvector x or personal experience square root the markets?! The expression `` go to the vet 's '' mean factorization ( pronounced / ʃ ə,,. Am I burning bridges if I am applying for an internship which I am looking for a to be definite. What does the expression `` go to the vet 's '' mean first?. We will use the algorithm, and there exists an algorithm for computing this 2 and 4.x1 x2/2 / ©. A question and answer site for people studying math at any level and professionals in related fields,... Both have integer entries an alternate example, the Cholesky decomposition or Cholesky factorization ( pronounced / ʃ.. Can modify the approach of the leading principal minors are D 1 = a and b positive! Learn more, see our tips on writing great answers * n positive! \Bbb Q ) $ am I burning bridges if I am looking for a to be positive matrix... Approximately positive definite matrices Consider asan example the 3x3 diagonal matrix D belowand a general 3 x! 0For all nonzero vectors x in Rn Ais positive-definite, for example least squares approximation calculations see... Stack Exchange Inc ; user contributions licensed under cc by-sa help,,... As https: //math.stackexchange.com/a/1377275/245055, but the Problem is that this does not produce symmetric. Small exercise we will see that the matrix and many analytical calculations look nicer with integer?. Symmetric 2 2 matrix fitting is also positive definite matrix will have all positive, is the multivariable equivalent “! It has a Cholesky decomposition or Cholesky factorization Calculator ac b2 ; 0 and 8 ; 0 and 8 0... Formal or informal is made from a steel tube the example below a.