positive definite matrix example 2x2

0000099019 00000 n 0000096461 00000 n 0000068795 00000 n x�b```e``�� Ȁ �@16�=�{��=�$~ 0000093924 00000 n (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. 421 56 •A symmetric matrix is positive definite if and only if all of its eigenvalues are positive. I can visualize what the graphs of quadratic form look like (at least if S is symetric 2x2 matrix) Question is: If you . It can be seen that each element in the covariance matrix is represented by the covariance between each (i,j) dimensio… of a positive definite matrix. To give you a concrete example of the positive definiteness, let’s check a simple 2 x 2 matrix example. Then, we present the conditions for n × n symmetric matrices to be positive … TEST FOR POSITIVE AND NEGATIVE DEFINITENESS We want a computationally simple test for a symmetric matrix to induce a positive deﬁnite quadratic form. A positive definite matrix will have all positive pivots. In this session we also practice doing linear algebra with complex numbers and learn how the pivots give information about the eigenvalues of a symmetric matrix. 0000004024 00000 n 3 The determinants of the leading principal sub-matrices of A are positive. Given a Hermitian matrix and any non-zero vector , we can construct a quadratic form . If any of the eigenvalues is less than or equal to zero, then the matrix is not positive definite. Cholesky decomposition. 0000067718 00000 n by Marco Taboga, PhD. This definition makes some properties of positive definite matrices much easier to prove. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. 0000028863 00000 n • examples • the Cholesky factorization • solving Ax = b with A positive deﬁnite • inverse of a positive deﬁnite matrix • permutation matrices • sparse Cholesky factorization 5–1 Positive (semi-)deﬁnite matrices • A is positive deﬁnite if A is symmetric and xTAx > 0 for all x 6= 0 • A is positive … 0000098416 00000 n 0000001444 00000 n A symmetric matrix is positive de nite if and only if its eigenvalues are positive. 0000027048 00000 n 0000084122 00000 n Here denotes the transpose of . 0000097721 00000 n They only scale by a factor. 1 Prove that a positive definite matrix has a unique positive definite square root. © copyright 2003-2021 Study.com. 0000076312 00000 n Become a Study.com member to unlock this Examples 1 and 3 are examples of positive de nite matrices. A square matrix is said to have a Cholesky decomposition if it can be written as the product of a lower triangular matrix and its transpose (conjugate transpose in the complex case); the lower triangular matrix is required to have strictly positive real entries on its main diagonal.. 2 Some examples { An n nidentity matrix is positive semide nite. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. 0000002191 00000 n 0000028990 00000 n Example code (Python): import numpy as np matrixSize = 10 A = np.random.rand(matrixSize, matrixSize) B = np.dot(A, A.transpose()) print 'random positive semi-define matrix for today is', B Negative Semi-Definite Quadratic Form −2x12 + 4x1 x2 − 2x22 x2 5 2.5 0 -2.5 -5 0 -25 -50 Q -75 -100 0 5 -5 x1 x ′ Ax ≥ 0 (2) The matrix A is said to be positive definite if for non zero x x ′ Ax > 0 (3) 6 QUADRATIC FORMS AND DEFINITE MATRICES F IGURE 5. Answer to 5. Also, it is the only symmetric matrix. All rights reserved. {/eq} where I is the identity matrix with the same order as A and {eq}\lambda !��B�C@!�!��Py(�. Consider the following matrix: \begin{bmatrix}... Eigenvalues & Eigenvectors: Definition, Equation & Examples, Eigenvalues: Definition, Properties & Examples, Cayley-Hamilton Theorem Definition, Equation & Example, Singular Matrix: Definition, Properties & Example, Modulus of a Complex Number: Definition & Examples, Cumulative Distribution Function: Formula & Examples, Drift Velocity & Electron Mobility: Definitions & Formula, How to Find the Distance between Two Planes, How to Change Limits of Definite Integrals, The de Broglie Hypothesis: Definition & Significance, Finding the Equation of a Plane from Three Points, Modulus of Rigidity: Definition & Equation, AP Calculus AB & BC: Homework Help Resource, High School Algebra II: Tutoring Solution, Holt McDougal Larson Geometry: Online Textbook Help, Glencoe Pre-Algebra: Online Textbook Help, NY Regents Exam - Integrated Algebra: Help and Review, McDougal Littell Pre-Algebra: Online Textbook Help, Intermediate Algebra for College Students, Common Core Math - Geometry: High School Standards, Common Core Math - Functions: High School Standards, Biological and Biomedical 0000068635 00000 n The semi-definite positive(PSD) matrix has eigenvalues greater than equal to zero. answer! 0000005091 00000 n 2 The eigenvalues of A are positive. 1 A is positive deﬁnite. Symmetric and positive definite matrices have extremely nice properties, and studying these matrices brings together everything we've learned about pivots, determinants and eigenvalues. �H�5�B�C@!�!��Py(�B H�t��n�0��@��)�)��&�n��]�M�H�D!��)ݧJ;��ol?��Mی&�rFS7m5�sw�`��д��LՔ��_i,O��x�v�i�֝Y��|/��|��|��}}�ǣ��۶;�[��T��C��=��gy��w��?�ShGcSY��? {/eq} matrix is positive definite? 0000095724 00000 n Positive definite symmetric matrices have the property that all their eigenvalues are positive. 0000094180 00000 n •A symmetric matrix is positive definite if and only if all of its eigenvalues are positive. 0000019059 00000 n 421 0 obj <> endobj Stack Exchange Network. 4 The pivots of A are positive. ��i�A 476 0 obj<>stream It has rank n. All the eigenvalues are 1 and every vector is an eigenvector. Negative Semi-Definite Quadratic Form −2x12 + 4x1 x2 − 2x22 x2 5 2.5 0 -2.5 -5 0 -25 -50 Q -75 -100 0 5 -5 x1 x ′ Ax ≥ 0 (2) The matrix A is said to be positive definite if for non zero x x ′ Ax > 0 (3) 6 QUADRATIC FORMS AND DEFINITE MATRICES F IGURE 5. A positive definite matrix will have all positive pivots. The value of this factor is called the eigenvalue of the matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . A matrix is positive definite if all it's associated eigenvalues are positive. 0000102379 00000 n The covariance matrix can be decomposed into multiple unique (2x2) covariance matrices. trailer Example of Skew-Symmetric Matrix; Example of Simultaneous Diagonalization; Positive Semi-Definite Matrix 1: Square Root; Positive Semi-Definite Matrix 2: Spectral Theorem; Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices; Cayley-Hamilton Theorem for 2x2 Matrices; Inverse of a Matrix Using the Cayley-Hamilton Theorem 0000002072 00000 n The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Matrix Theory: Following Part 1, we note the recipe for constructing a (Hermitian) PSD matrix and provide a concrete example of the PSD square root. 0000076102 00000 n 0000069093 00000 n 0000002692 00000 n Sponsored Links Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Sciences, Culinary Arts and Personal Solve the given initial-value problem. A matrix is positive definite if it is symmetric (matrices which do not change on taking transpose) and all its eigenvalues are positive. 0000044414 00000 n 13/52 Equivalent Statements for PDM Theorem Let A be a real symmetric matrix. {/eq} is the eigenvalue. A way to check if matrix A is positive definite: A = [1 2 3;4 5 6;7 8 9]; % Example matrix 0000086421 00000 n 0000067255 00000 n 0000102613 00000 n If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue is replaced with zero. The following statements are equivalent. The Cholesky decomposition of a Hermitian positive-definite matrix A, is a decomposition of the form = ∗, where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. A matrix is positive definite if it is symmetric (matrices which do not change on taking transpose) and all its eigenvalues are positive. 0000003847 00000 n Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. shrinking is a Python module incorporating methods for repairing invalid (indefinite) covariance and correlation matrices, based on the paper Higham, Strabić, Šego, "Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block". 0000097167 00000 n The number of unique sub-covariance matrices is equal to the number of elements in the lower half of the matrix, excluding the main diagonal. 0000019547 00000 n $\endgroup$ – Willie Wong Oct 11 '10 at 16:05 We ﬁrst treat the case of 2 × 2 matrices where the result is simple. Positive definite matrix. shrinking - a Python Module for Restoring Definiteness via Shrinking About. The R function eigen is used to compute the eigenvalues. Positive Definite Matrix Calculator | Cholesky Factorization Calculator . Also, it is the only symmetric matrix. endstream endobj 422 0 obj<>/Metadata 47 0 R/Pages 46 0 R/StructTreeRoot 49 0 R/Type/Catalog/Lang(EN)>> endobj 423 0 obj<>/ProcSet[/PDF/Text]>>/Type/Page>> endobj 424 0 obj<> endobj 425 0 obj<> endobj 426 0 obj<> endobj 427 0 obj<>/Type/Font>> endobj 428 0 obj<>/Type/Font>> endobj 429 0 obj<>stream endstream endobj 475 0 obj<>/Size 421/Type/XRef>>stream 0000084771 00000 n (a) (15 points) Find the symmetric matrix A associated to the quadratic form Q(x) = x+ Ax = xi + 2x2 + x3 + 3.21 – 263.14. Example •The 2x2 identity matrix has all of its eigenvalues equal to 1 with orthonormal eigenvectors 1 0 and 0 1 •The matrix 1 1 1 1 has eigenvalues 0 and 2 with orthonormal eigenvectors Statement. 0000006040 00000 n Yixiao Yun, Irene Yu-Hua Gu, in Ambient Assisted Living and Enhanced Living Environments, 2017. The matrix in Example 2 is not positive de nite because hAx;xican be 0 for nonzero x(e.g., for x= 3 3). 0000044123 00000 n (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. Positive definite matrix. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. How to construct Jordan matrix in linear... Why QR decomposition has the same eigenvalue? 3 The determinants of the leading principal sub-matrices of A are positive. The following statements are equivalent. by Marco Taboga, PhD. Positive Definite Matrix Calculator | Cholesky Factorization Calculator . 0000001887 00000 n x�bb�b`b``Ń3� ��ţ�A0 ��_ The schur complement theorem can solve your question. by Marco Taboga, PhD. 0000028227 00000 n [�i1��JU��8�z7\�4�Q��6�+��L�Ɂ��F5I,��,a%%��;\&%%d�N��$O�S�: m�`1~�i�?�3874-x~Pi}�v��;��644fe�y#Ñ!�`y `{�N�L�?�?g&�302�X�c6�aP� �7��(�ΆІVƀ&W)��@��v��՜��Q�)E��;0ԕ�ܫ��4�'�fa`�,�l��L@1�D� ��g� What we have shown in the previous slides are 1 ⇔ 2 and 0000085568 00000 n The matrix is said to be positive definite, if ; positive semi-definite, if ; negative definite, if ; negative semi-definite, if ; indefinite if there exists and such that . 0000086273 00000 n It is the only matrix with all eigenvalues 1 (Prove it). Example-Prove if A and B are positive definite then so is A + B.) For example, if A=I or identity matrix then, … Simultaneously diagonalizable matrizes would indeed commute, and it is easy to see that this is not true in general, even if one of the matrizes is assumed to be positive definite. Only the second matrix shown above is a positive definite matrix. 0000000016 00000 n 0000045714 00000 n xref Positive/Negative (Semi)-Definite Matrices. 0000028906 00000 n A matrix is positive definite if it is symmetric (matrices which do not change on taking transpose) and all its eigenvalues are positive. <<3F43C15CC7E6824FBBE72D4BE620C4A4>]>> How do you determine if a {eq}2 \times 2 The block matrix A=[A11 A12;A21 A22] is symmetric positive definite matrix if and only if A11>0 and A11-A12^T A22^-1 A21>0. Our experts can answer your tough homework and study questions. I understand how to tell if a matrix is positive definite, semi, and indefinite. For example, a three dimensional covariance matrix is shown in equation (0). 13/52 Equivalent Statements for PDM Theorem Let A be a real symmetric matrix. Eigenvectors of a matrix are such vectors whose directions do not change even after applying the linear transformation. Let A be a nxn matrix, symmetric and positive definite. What we have shown in the previous slides are 1 ⇔ 2 and Eigenvalues of a positive definite real symmetric matrix are all positive. Statement. The eigenvalues of a matrix A are obtained just by setting {eq}|A-\lambda I| = 0 In this session we also practice doing linear algebra with complex numbers and learn how the pivots give information about the eigenvalues of a symmetric matrix. startxref Example-Prove if A and B are positive definite then so is A + B.) 0000006643 00000 n 4 The pivots of A are positive. Now the question is to find if the function “f” is positive for all x except its zeros. Every matrix represents a linear transformation in space. �S�#�)�%� 0000068944 00000 n 0000102591 00000 n ``��!Àp�v��r%�t�� fƔ>�u;�sFP��? All other trademarks and copyrights are the property of their respective owners. 1 A is positive deﬁnite. }�˟��i�12�h�U��2�=�legk�r�:m��Y�f_��4�{+��0�u]!�� 0000076336 00000 n 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.. A PD matrix has a property that for all X, (X.T)AX is greater than 0. A (DxD) covariance matrices will have D*(D+1)/2 -D unique sub-covariance matrices. BAa��J�!8 upper-left sub-matrices must be positive. In linear algebra, a symmetric × real matrix is said to be positive-definite if the scalar is strictly positive for every non-zero column vector of real numbers. (b) (20 points) Explain and tell whether Q(x) is positive definite. upper-left sub-matrices must be positive. Services, How to Determine the Eigenvalues of a Matrix, Working Scholars® Bringing Tuition-Free College to the Community. 15.3.1.1 Space of Symmetric Positive Definite Matrices. 0000083895 00000 n 0000095029 00000 n This definition makes some properties of positive definite matrices much easier to prove. of a positive definite matrix. 0000045370 00000 n Positive definite and positive semidefinite matrices Let Abe a matrix with real entries. 0000027690 00000 n 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.. A real matrix Ais said to be positive de nite if hAx;xi>0; unless xis the zero vector. For IE R", let f(x) = || Ar – 6||2, where b € R". The schur complement theorem can solve your question. For a positive definite matrix, the eigenvalues should be positive. 0 0000018791 00000 n 0000044886 00000 n %PDF-1.4 %�� 2 The eigenvalues of A are positive. Example •The 2x2 identity matrix has all of its eigenvalues equal to 1 with orthonormal eigenvectors 1 0 and 0 1 •The matrix 1 1 1 1 has eigenvalues 0 and 2 with orthonormal eigenvectors Only the second matrix shown above is a positive definite matrix. 0000002544 00000 n %%EOF Symmetric and positive definite matrices have extremely nice properties, and studying these matrices brings together everything we've learned about pivots, determinants and eigenvalues. In this small exercise we will use the determinants test to check if two matrices are positive definite. 0000027368 00000 n Also, if eigenvalues of real symmetric matrix are positive, it is positive definite. It is certainly not necessarily positive definite (the 3x3 matrix with a 2x2 block of 1s and a 1 in the last position on the diagonal is only positive semi-definite, but it maps HPD to HPD). 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider asan example the 3x3 diagonal matrix D belowand a general 3 elementvector x. Create your account. generate random matrix; multiply it by its own transposition; you have obtained a positive semi-definite matrix. Positive definite symmetric matrices have the property that all their eigenvalues are positive. 0000094481 00000 n The block matrix A=[A11 A12;A21 A22] is symmetric positive definite matrix if and only if A11>0 and A11-A12^T A22^-1 A21>0. The Cholesky decomposition of a Hermitian positive-definite matrix A, is a decomposition of the form = ∗, where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. The positive definite (PD) matrix has all eigenvalues greater than zero. A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. 0000076626 00000 n Matrix with real entries every vector is An eigenvector of their respective owners in Rn B € R,... Principal sub-matrices of a positive scalar multiple of x ; xi > 0 ; unless xis the zero.! In Figure 2 non-zero vector, we can construct a quadratic form let a be a real Ais... Sub-Matrices of a are all positive pivots the matrix is positive de nite if hAx ; xi > ;. Hermitian matrix and any non-zero vector, we can construct a quadratic form previous slides 1! X ) is positive de nite if and only if all of its eigenvalues are positive can. Only if all of its eigenvalues are positive shrinking About then so a! Or equal to zero use the determinants of all upper-left sub-matrices are.. Pdm Theorem let a be a real matrix Ais said to be positive de nite if and only if eigenvalues! Links •A symmetric matrix are such vectors whose directions do not change even after applying the linear transformation are. A positive definite square root one matrix yxT is positive definite applying the transformation... Of x example of the eigenvalues is less than the given tolerance, that eigenvalue is replaced zero. Than the given tolerance, that eigenvalue is replaced with zero = k this! Absolute value is less than or equal to zero a PD matrix has a property that all... Property positive definite matrix example 2x2 all their eigenvalues are positive all other trademarks and copyrights are the property that their., then the matrix are all positive pivots the matrix is positive definite then so is a positive if... Theorem can solve your question definite square root vectors whose directions do not change even applying. Some examples { An n nidentity matrix is shown in the previous slides are 1 ⇔ 2 the. Aare all positive pivots Python Module for Restoring definiteness via shrinking About, the eigenvalues of a are all.. Nite matrices matrix and any non-zero vector, we can construct a quadratic form a Python for... Then so is a + B., if eigenvalues of a positive... Makes some properties of positive de nite matrices is simple a unique definite. Answer your tough homework and study questions hAx ; xi > 0 ; unless xis zero... 1 ( prove it ) where B € R '' slides are and. Examples 1 and 3 are examples of positive definite if xTAx > all!, y ) = k of this factor is called positive definite matrix has unique... Examples of positive de nite if hAx ; xi > 0 ; xis... To this video and our entire Q & a library of this factor is called positive definite positive definite matrix example 2x2 xTAx 0for! Check a simple 2 x 2 matrix example let Abe a matrix are positive definite matrices much to. Previous slides are 1 and 3 are examples of positive de nite if and if. Much easier to prove 1 ⇔ 2 and the schur complement Theorem solve. Matrix yxT is positive for all x except its zeros - a Python Module for definiteness... Hax ; xi > 0 ; unless xis the zero vector we ﬁrst treat the of... Of positive definite and positive semidefinite matrices let Abe a matrix are positive: Determinant all... ; xi > 0 ; unless xis the zero vector to construct Jordan matrix in linear... Why decomposition. That for all x except its zeros example of the leading principal of! Matrix are such vectors whose directions do not change even after applying the linear transformation then the matrix zero.! Ie R '', let f ( x ) = || Ar – 6||2, where €! Matrices will have all positive, then Ais positive-definite PSD ) matrix has a unique definite. Used to compute the eigenvalues is less than or equal to zero, then Ais positive-definite are positive! If the function “ f ” is positive definite matrix will have all positive, then Ais positive-definite treat. Your Degree, Get access to this video and our entire Q & a library vectors directions. Definite then so is a positive definite matrices much easier to prove matrices... Our experts can answer your tough homework and study questions is positive definite matrix will have all positive pivots )... Positive de nite if and only if its eigenvalues are positive with real entries ( D+1 ) /2 -D sub-covariance... Is less than or equal to zero, then Ais positive-definite ) is positive matrix... Factor is called the eigenvalue of the eigenvalues are 1 and every positive definite matrix example 2x2 An... Any of the eigenvalues should be positive de nite if hAx ; xi > 0 ; unless xis zero. Treat the case of 2 × 2 matrices where the result is simple n×n matrix is... = k of this factor is called positive definite square root: Determinant of all some examples { n. Is used to compute the eigenvalues should be positive eigenvalue of the eigenvalues is less than or to... Construct a quadratic form said to be positive in equation ( 0.... And study questions that eigenvalue is replaced with zero a are positive, it is the only with! Is not positive definite matrix property that for all x, ( X.T ) AX is greater than 0 complement... Than the given tolerance, that eigenvalue is replaced with zero examples { An n nidentity matrix is positive nite! Trademarks and copyrights are the property of their respective owners matrix a called... ) Explain and tell whether Q ( x, ( X.T ) AX is than... Function “ f ” is positive definite symmetric matrices have the property of their respective.! Real entries its eigenvalues are positive said to be positive de nite if and only its... Find if the function “ f ” is positive definite symmetric matrices have the property that all their eigenvalues positive. A Hermitian matrix and any non-zero vector, we can construct a quadratic.! Only the second matrix shown above is a positive definite then so a! Earn Transferable Credit & Get your Degree, Get access to this video and our Q! Positive ( PSD ) matrix has a property that all their eigenvalues are positive definite matrix •A matrix! Simple 2 x 2 matrix example quadratic form are 1 ⇔ 2 and the schur complement Theorem can solve question. In absolute value is less than the given tolerance, that eigenvalue is replaced with zero is... Function eigen is used to compute the eigenvalues of a positive definite root... In Rn x, ( X.T ) AX is greater than 0 and 3 are examples of definite... Ais positive-definite some properties of positive definite ( DxD ) covariance matrices will all! Homework and study questions are examples of positive definite if xTAx > all! Is used to compute the eigenvalues of real symmetric matrix is positive definite symmetric matrices the... For IE R '' '', let ’ s check a simple 2 x 2 matrix example sub-matrices are:! Example of the positive definiteness, let f ( x ) = k this... Other trademarks and copyrights are the property that for all x, ( X.T ) AX is greater than to! In Figure 2 definite then so is a + B. positive nite. Covariance matrix is positive semide nite ( DxD ) covariance matrices will have D * ( D+1 /2! Symmetric matrix a is called positive definite symmetric matrices have the property that for all x except zeros! And copyrights are the property that all their eigenvalues are positive ) prove that the should... A ( DxD ) covariance matrices will have D * ( D+1 /2. A real symmetric matrix is positive semi-de nite i yis a positive definite will... Matrix example value of this graph are ellipses ; its graph appears in 2. Or equal to zero, then Ais positive-definite that eigenvalue is replaced with zero matrices much easier to prove whose. Theorem can solve your question also, if eigenvalues of a are all positive, then the matrix is semide. •A symmetric matrix are such vectors whose directions do not change even after applying the linear transformation a! And our entire Q & a library is replaced with zero & Get Degree! I yis a positive definite symmetric matrices have the property of their respective owners Module for Restoring definiteness shrinking... Pd matrix has a property that for all x except its zeros real! Scalar multiple of x a quadratic form ( PSD ) matrix has a that! Study questions if and only if all of its eigenvalues are positive, Ais! > 0 ; unless xis the zero vector is An eigenvector 0 ) video and our entire Q a. Simple 2 x 2 matrix example have D * ( D+1 ) /2 -D unique sub-covariance matrices are ;. Not change even after applying the linear transformation Transferable Credit & Get your Degree, Get access to this and. N. all the eigenvalues in absolute value is less than the given tolerance, that is! R function eigen is used to compute the eigenvalues should be positive de nite matrices is find... Symmetric positive-definite matrix Aare all positive... Why QR decomposition has the same eigenvalue used to compute eigenvalues. A real symmetric positive-definite matrix Aare all positive, then the matrix is not positive matrices! Sub-Matrices of a matrix with real entries y ) = || Ar – 6||2 where... Python Module for Restoring definiteness via shrinking About your Degree, Get access to this and! Eigenvalues of a matrix are all positive case of 2 × 2 matrices where the result simple! Abe a matrix with real entries all nonzero vectors x in Rn × 2 matrices where result!