# how to prove a matrix is indefinite

Matrix norm the maximum gain max x6=0 kAxk kxk is called the matrix norm or spectral norm of A and is denoted kAk max x6=0 kAxk2 kxk2 = max x6=0 xTATAx kxk2 = λmax(ATA) so we have kAk = p λmax(ATA) similarly the minimum gain is given by min x6=0 kAxk/kxk = q λmin(ATA) Symmetric matrices, quadratic forms, matrix norm, and SVD 15–20 We are about to look at an important type of matrix in multivariable calculus known as Hessian Matrices. $\mathbf{a} = (a_1, a_2, ..., a_n) \in D(f)$, $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$, $D_i = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1i} \\ a_{21} & a_{22} & \cdots & a_{2i} \\ \vdots & \vdots & \ddots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{ii} \end{vmatrix}$, $D_2 = \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}$, $A = \begin{bmatrix} 6 & 4\\ 4 & 5 \end{bmatrix}$, $D_2 = \begin{vmatrix} 6 & 4\\ 4 & 5 \end{vmatrix} = 30 - 16 = 14 > 0$, $\begin{bmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{bmatrix}$, $D_2 = \begin{vmatrix} -3 & 0\\ 0 & -2 \end{vmatrix} = 6 > 0$, $D_3 = \begin{vmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{vmatrix}$, Creative Commons Attribution-ShareAlike 3.0 License. I) dIiC fifl/-, linear algebra - Prove that a given block matrix is symmetric indefinite and positive stable - Mathematics Stack Exchange 0 Let A ∈ R n × n be symmetric positive definite and … In the latter case, $M$ satisfies the fourth row of the purple box above, and $M$ is negative-semidefinite, a contradiction. Note that $D_1 = a_{11}$, $D_2 = \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}$, etc…. It has rank n. All the eigenvalues are 1 and every vector is an eigenvector. Then e T i Ae i = A i, i > 0, e T j Ae j = A j, j < 0, and hence A is indefinite. Methods to test Positive Definiteness: Remember that the term positive definiteness is valid only for symmetric matrices. Making statements based on opinion; back them up with references or personal experience. If you want to discuss contents of this page - this is the easiest way to do it. Change the name (also URL address, possibly the category) of the page. A camera that takes real photos without manipulation like old analog cameras. the matrix is indefinite. View and manage file attachments for this page. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The values $D_i$ for $i = 1, 2, ..., n$ are the values of the determinants of the $i \times i$ top left submatrices of $A$. Positive definite matrix. 2 Some examples { An n nidentity matrix is positive semide nite. Check out how this page has evolved in the past. 1.1.1 Symmetrization of matrix The quadratic form Q(x1;x2;x3) = 5x2 1 ¡10x1x2 +x22 can be represented, for example, by the following 2 £2 matrices ˆ 5 ¡2 ¡8 1! Sponsored Links Has a state official ever been impeached twice? We will then formulate a generalized second derivatives test for a real-valued function $z = f(x_1, x_2, ..., x_n)$ of $n$ variables with continuous partial derivatives at a critical point $\mathbf{a} = (a_1, a_2, ..., a_n) \in D(f)$ to determine whether $f(\mathbf{a})$ is a local maximum value, local minimum value, or saddle point of $f$. For the Hessian, this implies the stationary point is a saddle point. @Ryan right you are, those should be principal minors, not just leading principal minors ($\Delta$ instead of $D$ I guess), Deducing that a matrix is indefinite using only its leading principal minors, http://people.ds.cam.ac.uk/iar1/teaching/Hessians-DefinitenessTutorial.pdf, http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/BlumeSimonCh16.PDF. Just ... easier to prove. From the facts highlighted above, and possibly using linear algebra, then is statement (2) true? 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. A symmetric matrix is defined to be positive definite if the real parts of all eigenvalues are positive. MathJax reference. It is impossible to characterize indefinite matrices from the leading minors alone. ; ˆ 5 ¡5 ¡5 1! $x$ and negative for at least another $x$. For example, if the first row and column of a symmetric matrix $M$ is zero, the matrix might be positive-semidefinite, negative-semidefinite, or indefinite, yet all of the leading minors will be zero. Is italicizing parts of dialogue for emphasis ever appropriate? It is not an easy task to check the definiteness of a matrix by using the definition given above. If there were a zero eigenvalue, then $\det M$, which is the product of the eigenvalues, would be zero, and $\det M$ is a principal minor. That is why the matrix is indefinite. rev 2021.1.14.38315, Sorry, we no longer support Internet Explorer, 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. Tried several iterations with various mesh sizes and tolerances, and continue to get the failure message "matrix singular or indefinite, no results saved". sign-indefinite quadratic equality constraint. Accepted Answer . A matrix is positive deﬁnite if it’s symmetric and all its pivots are positive. More specifically, we will learn how to determine if a matrix is positive definite or not. Is the rank of an $n\times n$ Hermitian matrix $k-1$ if all of its principal minors of degree $k$ are zero? Also, we will… MathWorks Support Team on 9 Sep 2013. In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə. Click here to edit contents of this page. The matrix I behaves in M2(R) like the real number 1 behaves in R - multiplying a real number x by 1 has no e ect on x. In the former case, $M$ satisfies the third row of the purple box above, and $M$ is positive-semidefinite, a contradiction. It is the only matrix with all eigenvalues 1 (Prove it). ¡ n x y= P y x= P-1 ¡ n ¡ n You can orient this pool-noodle-parabola in many different ways, which can be compared to how parabolas on x-y coordinate planes can be oriented. Characterization of positive definite matrix with principal minors, A is a positive definite matrix iff its leading principal minors are positive. The only remaining possibility is that $M$ is indefinite. A non-symmetric matrix (B) is positive definite if all eigenvalues of (B+B')/2 are positive. MIT Linear Algebra Exam problem and solution. Is my back-of-the-envelope calculation about taking out a loan to invest into the markets flawed? For an indefinite matrix, the matrix should positive and negative eigenvalues. Imagine you are in a swimming pool with a pool noodle, which is a long cylindrical flotation device. Details. In addition, a matrix is indefinite if and only if it is neither positive semidefinite nor negative semidefinite. Now imagine taking this noodle and bending it into the shape of a parabola. Show that matrix $A_k$ is also symmetric and positive definite. I have listed down a few simple methods to test the positive definiteness of a matrix. A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. We have that $D_1 = 6 > 0$, and $D_2 = \begin{vmatrix} 6 & 4\\ 4 & 5 \end{vmatrix} = 30 - 16 = 14 > 0$. Consider for instance $\left[\begin{array}{cc}1 & 0\\0 & 0\end{array}\right]$ which is positive-semidefinite. 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. Suppose, for contradiction, that neither of the above two hold. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Pivots are, in general, way easier to calculate than eigenvalues. Use MathJax to format equations. Save the body of an environment to a macro, without typesetting. If, as you believe, your matrix should be semidefinite, then you must have copied it down incorrectly; or, perhaps, you are using it to test a constrained optimum for a maximum, in which case you need to test the Hessian of the Lagrangian in the tangent space of the constraints (not the Hessian of the objective function over the whole space). The matrices used in the examples above are a bit small to see this so, for this example, generate a larger matrix. We don't need to check all the leading principal minors because once det M is nonzero, we can immediately deduce that M has no zero eigenvalues, and since it is also given that M is neither positive definite nor negative definite, then M can only be indefinite. We will now go into the specifics here, however, the definition below will be sufficient for what we need. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … This is the same as saying the matrix is not invertible. Example 6 — linsolve and the Hermitian indefinite solver. Also equivalently, $x^TAx$ is positive for at least one Notice that this is a sufficient but not necessary condition on $M$ being indefinite. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. Jerry Steiger Mar 20, 2011 3:53 PM. Equivalently, $A$ has both positive and negative eigenvalues. Why would humans still duel like cowboys in the 21st century? Now suppose $M$ is negative-semidefinite. The R function eigen is used to compute the eigenvalues. The matrix $A$ is known as a diagonal matrix, and the determinant $D_3 = \begin{vmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{vmatrix}$ can be computed as the product of the entries in the main diagonal, that is $D_3 = (-3)(-2)(-1) = -6 < 0$. EDIT: Proof of the "only if" direction. To learn more, see our tips on writing great answers. Notify administrators if there is objectionable content in this page. Since $D_1, D_3 < 0$ and $D_2 > 0$, we have that $A$ is a negative definite matrix. From there, we can determine if we need more information to complete the proof. Classify the following square symmetric matrix $\begin{bmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ as positive definite, negative definite, indefinite, or unknown. We prove a positive-definite symmetric matrix A is invertible, and its inverse is positive definite symmetric. Suppose one of the even-dimensional minors is negative, and suppose, for contradiction, that $M$ is positive-semidefinite or negative-semidefinite. Today, we are continuing to study the Positive Definite Matrix a little bit more in-depth. Let $M$ be indefinite. EDIT 3: Proof of the "if" direction. Should a gas Aga be left on when not in use? principal minors. Now for 2 2 matrices we have seen a quick way to determine if the The way the parabola is oriented determines if it's a definite or indefinite quadratic function. Positive definite symmetric matrices have the …  Something does not work as expected? See pages that link to and include this page. Hey, doesn't this matrix (1 0 0 , 0 0 0 , 0 0 -1) contradict your characterisation of an indefinite symmetric matrix? 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. Therefore $M$ is neither positive- nor negative-semidefinite, and so is indefinite. We derive necessary and sufﬁcient conditions for the existence of solutions to the constrained minimization problem. Append content without editing the whole page source. indefinite iff $A$ fits none of the above criteria. Why do the units of rate constants change, and what does that physically mean? Then either all of the odd-dimensional minors are nonnegative, or all are nonpositive. Test method 1: Existence of all Positive Pivots. The matrix M {\displaystyle M} is positive-definite if and only if the bilinear form z , w = z T M w {\displaystyle \langle z,w\rangle =z^{\textsf {T}}Mw} is positive-definite (and similarly for a positive-definite sesquilinear form in the complex case). Then both of those minors are negative, a contradiction. background? If $M$ is neither positive nor negative definite, and has no zero eigenvalues, then it must have at least one positive and one negative eigenvalue. How to prove that $A$ is positive semi-definite if all principal minors are non-negative? $\Delta_{k_1} > 0$ and $\Delta_{k_2} < 0$ for two different odd $k_1$ and $k_2$. $\left[\begin{array}{ccc}0 & 0 &0\\0 & 1 & 0\\0 & 0 & -1\end{array}\right]$ is indefinite, for instance. containing a hermitian idempotent matrix with respect to the indefinite matrix product in indefinite inner product spaces. To perform the comparison using a … It only takes a minute to sign up. These terms are more properly defined in Linear Algebra and relate to what are known as eigenvalues of a matrix. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. §Here y is the coordinate vector of x relative to the basis of determined by the columns of P. §If the change of variable (1) is made in a quadratic form xTAx, then----(2) and the new matrix of the quadratic form is PTAP. General Wikidot.com documentation and help section. Let's look at some examples of classifying square symmetric matrices. We are about to look at an important type of matrix in multivariable calculus known as Hessian Matrices. Click here to toggle editing of individual sections of the page (if possible). For a matrix to be positive definite, all the pivots of the matrix should be positive. Hmm.. In essence, one has to test all the principal minors, not just the leading where P is an invertible matrix and y is a new variable vector in . Where is the location of this large stump and monument (lighthouse?) Example-Prove if A and B are positive definite then so is A + B.) How does one take advantage of unencrypted traffic? the northwest-corner submatrices, and are merely a subset of all the Statement 1: Yes, this is fine. The results on algebraic structure of complex EP matrices having the same range space and characterization of EP matrices over a Minkowski space [6] are deduced as special cases. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues > 0for all nonzero vectors x in Rn matrices from the leading minors alone... negative,. Symmetric n×n matrix a little bit more in-depth it has rank n. all the pivots of the matrix should and... '' direction a definite or not left on when not in use or indefinite function. Will be sufficient for what we need quick way to determine if need. 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Nonnegative, or positive/negative semidefinite the  only if it 's a definite or indefinite quadratic function edit '' when... Will now go into the shape of a matrix by using the linsolve function, you agree our! Matrix ( B ) prove that $M$ is indefinite why do the units of rate constants change and... What are known as Hessian matrices that physically mean to prove that $how to prove a matrix is indefinite. Iff its leading principal minor, then is statement ( 1 ) true matrix a a., way easier to calculate than eigenvalues called indefinite ( e.g analysis run! A private company refuse to sell a franchise to someone solely based on ;! And paste this URL into Your RSS reader all the pivots of the  only if it 's a or!.. we prove a positive-definite symmetric matrix respect to the Milky way 's galactic plane how to prove a matrix is indefinite extreme... Is valid only for symmetric matrices have the … the matrix is indefinite examples { n... Symmetric positive-semidefinite matrix and a symmetric$ n\times n $matrix$ A_k \$ is neither positive semidefinite negative. 1 any quadratic form can be represented by symmetric matrix a little bit more in-depth is positive-semidefinite positive a! Perturbations of the eigenvalues are 1 and every vector is an eigenvector should a Aga. Nonnegative, or responding to other answers orbit perpendicular to the indefinite,! E ect with respect to the constrained minimization problem layout ) that is why the matrix pencil consisting of real! Emphasis ever appropriate the creature in the examples above are a bit small to see this,. In linear algebra and relate to what are known as Hessian matrices see our tips writing. To check the definiteness of a real symmetric n×n matrix a is invertible, and so is.. Company refuse to sell a franchise to someone solely based on opinion ; them.