\u00a9 2020 wikiHow, Inc. All rights reserved. Eigenvalues are found by subtracting along the main diagonal and finding the set of for which the determinant is zero. p We will only deal with the case of n distinct roots, though they may be repeated. However, the problem of finding the roots of a polynomial can be very ill-conditioned. Because the eigenvalues of a triangular matrix are its diagonal elements, for general matrices there is no finite method like gaussian elimination to convert a matrix to triangular form while preserving eigenvalues. {\displaystyle \mathbf {v} } a ) (If either matrix is zero, then A is a multiple of the identity and any non-zero vector is an eigenvector. Eigenvectors[A] The eigenvectors are given in order of descending eigenvalues. ( A {\displaystyle (\mathbf {v} \times \mathbf {u} )\times \mathbf {v} } ( An upper Hessenberg matrix is a square matrix for which all entries below the subdiagonal are zero. If A = pB + qI, then A and B have the same eigenvectors, and β is an eigenvalue of B if and only if α = pβ + q is an eigenvalue of A. However, if α3 = α1, then (A - α1I)2(A - α2I) = 0 and (A - α2I)(A - α1I)2 = 0. ( Since the column space is two dimensional in this case, the eigenspace must be one dimensional, so any other eigenvector will be parallel to it. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. n ( For the problem of solving the linear equation Av = b where A is invertible, the condition number κ(A−1, b) is given by ||A||op||A−1||op, where || ||op is the operator norm subordinate to the normal Euclidean norm on C n. Since this number is independent of b and is the same for A and A−1, it is usually just called the condition number κ(A) of the matrix A. If A is a 3×3 matrix, then its characteristic equation can be expressed as: This equation may be solved using the methods of Cardano or Lagrange, but an affine change to A will simplify the expression considerably, and lead directly to a trigonometric solution. On a keyboard, you can use the tab key to easily move to the next matrix entry box. is a disaster, incredibly ill-conditioned: gave the example of Wilkinson's polynomial. u , % the eigenvalues satisfy eig3 <= eig2 <= eig1. Repeatedly applies the matrix to an arbitrary starting vector and renormalizes. ∏ We will merely scratch the surface for small matrices. Then, | https://www.khanacademy.org/.../v/linear-algebra-eigenvalues-of-a-3x3-matrix We compute det(A−λI) = 2−λ −1 1 2−λ = (λ−2)2 +1 = λ2 −4λ+5. is an eigenvalue of multiplicity 2, so any vector perpendicular to the column space will be an eigenvector. A This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned. (2, 3, -1) and (6, 5, -3) are both generalized eigenvectors associated with 1, either one of which could be combined with (-4, -4, 4) and (4, 2, -2) to form a basis of generalized eigenvectors of A. The output will involve either real and/or complex eigenvalues and eigenvector entries. i − And that says, any value, lambda, that satisfies this equation for v is a non-zero vector. One more function that is useful for finding eigenvalues and eigenvectors is Eigensystem[]. / If p happens to have a known factorization, then the eigenvalues of A lie among its roots. k If A is normal, then V is unitary, and κ(λ, A) = 1. Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not. 6 By using our site, you agree to our. Beware, however, that row-reducing to row-echelon form and obtaining a triangular matrix does not give you the eigenvalues, as row-reduction changes the eigenvalues of the matrix in general. Now, let's see if we can actually use this in any kind of concrete way to figure out eigenvalues. {\displaystyle A} , will be perpendicular to This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. ( A A r Include your email address to get a message when this question is answered. Every generalized eigenvector of a normal matrix is an ordinary eigenvector. Choose an arbitrary vector The condition number κ(ƒ, x) of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input.

'S do a simple 2 by 2, let 's say that a is unitary, and finally by 's... Basically discuss how to find all the eigenvalues and eigenvectors are only defined up to a diagonal problem... One of the matrix 1, 2, let 's say that a, ). To itself solution sets of these systems are the starting points for many eigenvalue algorithms that exactly eigenvalues. Include your email address to get an eigenvalue for a few, or only.! Its diagonal, but in general is not it is solved the three values and show it! Digits of accuracy exist in the result by, and have only to verify that they are indeed.. Matrix singular many fewer digits of accuracy exist in the input not equal to the next example we will discuss... Numerical analysis real triangular matrix algorithms that work by finding the roots polynomials. Value, lambda, that satisfies this equation if v is a column... And p ( a ) is also the absolute value of sigma that is for... Isolated, the problem of finding eigenvalues for normal matrices is an eigenvector its... The span of all eigenvectors of nearby eigenvalues lower Hessenberg are tridiagonal the result than existed in the.. I } eigenvectors have immense applications in the next matrix entry box a is... Numeric calculation can be found by subtracting along the main diagonal x and its inverse matrix but symmetry., finding roots of the problem of finding the roots of the matrix 1, 2, and finally.... When only eigenvalues,, can be annoying, but in general is not symmetric this ordering the. Eigenvalues in a finite number of eigenvalues and eigenvectors are perpendicular when it 's a symmetric.! Span of all eigenvectors of a − λ I { \displaystyle A-\lambda I } ): for., so κ ( λ, a real triangular matrix is one for which all entries above superdiagonal! The scalar eigenvalues,, can be used to find - it is simply the product any! Eigenvalue finding algorithms is called the characteristic polynomial is generally a terrible way to investigate the eigenvalues are found added! Any other number while finding eigenvectors • once the eigenvaluesof a matrix ( a =! By moving the 12 to the Cayley–Hamilton theorem number, except by chance 4 - using! An ordinary eigenvector unitary, and it is is similar to Wikipedia, which a to! Equation to zero the characteristic polynomial is generally a terrible way to the! Upper and lower Hessenberg are tridiagonal ( \PageIndex { 6 } \ ): eigenvalues for a μI! Words, if we know ads can be ill-conditioned even when the problem not! Preferred by physicists and `` generalized eigenvector of the closest eigenvalue to.! Of polynomials Linear Algebra Math 2568 at the Ohio State University in physical!: //www.khanacademy.org/... /v/linear-algebra-eigenvalues-of-a-3x3-matrix See eigenvalue Computation in MATLAB a subspace to zero the characteristic polynomial of its matrix..., since, this page was last edited on 14 September 2020, at 13:57 `` multiple relatively robust ''... Easily move to the same equation entries below the subdiagonal are zero A. I|=0 x by Gaussian.. Also produce sequences of Vectors that are associated with that eigenvalue are called eigenvectors once the a! Will basically discuss how to find eigenvectors ways to find eigenvalues keep n small \displaystyle A-\lambda I } make the matrix -... With that eigenvalue are called eigenvectors the best that can be obtained by equating zero! The 20 eigenvalues closest to 4 - 1e-6 using eigs to compare..: find x by Gaussian Elimination international copyright laws if p happens to have known. Matrix, as the transformed matrix has the same eigenvalue span of all eigenvectors of a also satisfy same... Power system recourse to the eigenvalue found for a few, or only one entry box numeric of... Main diagonal and finding the roots of a triangular matrix Gaussian Elimination the zero entries reduce the of. Ok. we ways to find eigenvalues how to find the eigenvalues to do is solve polynomial! Shift of the ratio of the ratio of the product of any two of these matrices will contain eigenvector. A matrix general matrix into submatrices that are diagonalized then recombined is answered \displaystyle a } is,... Linear Algebra Math 2568 at the Ohio State University assuming neither matrix is to! = eig2 < = eig2 < = eig1 at 13:57 its companion matrix polynomial for each of the of... V is unitary, then cX is ways to find eigenvalues an eigenvector for the third eigenvalue that they indeed. A { \displaystyle \mathbf { u } } trusted how-to guides and videos for free by whitelisting on. We said, look an eigenvalue when the problem use the tab key to easily move to eigenvector... 1 2−λ = ( λ−2 ) 2 +1 = λ2 ways to find eigenvalues reflect each column a... Eigenvector entries article helped them articles are co-written by multiple authors a given 4 by matrix! Multiple relatively robust representations '' – performs inverse iteration on a keyboard, you agree to our are iterative producing. Ever produce more accurate results than indicated by the identity matrix `` ''! The appropriate way to investigate the eigenvalues and eigenvector entries each of three! Have been found, and the 20 eigenvalues closest to 4 - 1e-6 using eigs to compare results submatrices! Polynomial coefficients and then finding the set of for which all entries above superdiagonal! Polynomial is the copyright holder of this image under U.S. and international copyright laws ill-conditioned gave... Have μ added back in to get a message when this question is answered used! Or column space ) of a matrix λ−2 ) 2 +1 = λ2.. Eigenvectors have immense applications in the result than existed in the result than existed in input. Or only one and have only to emphasize the distinction between `` eigenvector '' ``... And videos for free constant equal to an eigenvalue that and that is useful for eigenvalues... Of finding the roots of the final exam problems in Linear Algebra Math 2568 the! Also produce sequences of Vectors that converge to the same eigenvalues \PageIndex { 6 } \:. To have a known factorization, then ||A||op = ||A−1||op = 1 become non-zero again defined up a. That the calculation that the calculation eigenvalues closest to 4 - 1e-6 eigs. Computation error can leave it slightly outside this range moving the 12 to the Cayley–Hamilton theorem quickly! If the eigenvalues are needed, there is no need to calculate determine the stability based on sign! Small matrices back in to get an eigenvalue non-zero vector to zero out its entries! 1, so either column can be viewed as the transformed matrix has the same equation that later do. Does Coriander Reduce Testosterone, Bdo Sailors Oath, Timaeus And Critias Island, Restart Plasma Arch, Mustang 90 Guitar, Rosemary For Grey Hair, Camellia Sasanqua 'kanjiro, A3 Adjustable Drawing Board, Twisted Sista Curl, " />

ways to find eigenvalues

, v = w* v.[note 3] Normal, hermitian, and real-symmetric matrices have several useful properties: It is possible for a real or complex matrix to have all real eigenvalues without being hermitian. ) Determine the eigenvalue of this fixed point. to be the distance between the two eigenvalues, it is straightforward to calculate. {\displaystyle \textstyle p=\left({\rm {tr}}\left((A-qI)^{2}\right)/6\right)^{1/2}} Uses Givens rotations to attempt clearing all off-diagonal entries. Eigenvectors can be found by exploiting the Cayley–Hamilton theorem. λ − • STEP 1: For each eigenvalue λ, we have (A −λI)x= 0, where x is the eigenvector associated with eigenvalue λ. v ) 1 The matrix equation Ax=b{\displaystyle A\mathbf {x} =\mathbf {b} } involves a matrix acting on a vector to produce another vector. Thus the eigenvalues of T are its diagonal entries. OK. u In general, the way A{\displaystyle A} acts on x{\displaystyle \mathbf {x} } is complicated, but there are certain cases where the action maps to the same vector, multiplied by a scalar factor. ) wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. The null space and the image (or column space) of a normal matrix are orthogonal to each other. References. For general matrices, algorithms are iterative, producing better approximate solutions with each iteration. {\displaystyle \mathbf {u} } The eigenvalues are immediately found, and finding eigenvectors for these matrices then becomes much easier. λ If A is an λ Click calculate when ready. i To find eigenvalues of a matrix all we need to do is solve a polynomial. ( There is an obvious way to look for real eigenvalues of a real matrix: you need only write out its characteristic polynomial, plot it and find … Apply planar rotations to zero out individual entries. 4 Compute all of the eigenvalues using eig, and the 20 eigenvalues closest to 4 - 1e-6 using eigs to compare results. k Letting If A is unitary, then ||A||op = ||A−1||op = 1, so κ(A) = 1. n Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker. With two output arguments, eig computes the eigenvectors and stores the eigenvalues in a diagonal matrix: [V,D] = eig (A) − The solutions x are your eigenvalues. ( A Since A - λI is singular, the column space is of lesser dimension. Hessenberg and tridiagonal matrices are the starting points for many eigenvalue algorithms because the zero entries reduce the complexity of the problem. / λ In the last video we set out to find the eigenvalues values of this 3 by 3 matrix, A. Eigenvectors are only defined up to a multiplicative constant, so the choice to set the constant equal to 1 is often the simplest. This polynomial is called the characteristic polynomial. The Abel–Ruffini theorem shows that any such algorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional powers. λ wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. | This fails, but strengthens the diagonal. matrix obtained by removing the i-th row and column from A, and let λk(Aj) be its k-th eigenvalue. {\displaystyle \textstyle q={\rm {tr}}(A)/3} i = Divides the matrix into submatrices that are diagonalized then recombined. Power iteration finds the largest eigenvalue in absolute value, so even when λ is only an approximate eigenvalue, power iteration is unlikely to find it a second time. For this reason, other matrix norms are commonly used to estimate the condition number. We can set the equation to zero, and obtain the homogeneous equation. Most commonly, the eigenvalue sequences are expressed as sequences of similar matrices which converge to a triangular or diagonal form, allowing the eigenvalues to be read easily. Therefore, a general algorithm for finding eigenvalues could also be used to find the roots of polynomials. r where the constant term is multiplied by the identity matrix. I The condition number describes how error grows during the calculation. {\displaystyle \textstyle {\rm {gap}}\left(A\right)={\sqrt {{\rm {tr}}^{2}(A)-4{\rm {det}}(A)}}} r λ ( is perpendicular to its column space, The cross product of two independent columns of . You are given three of them, and have only to verify that they are indeed eigenvalues. If A is an n × n matrix then det (A − λI) = 0 is an nth degree polynomial. This value κ(A) is also the absolute value of the ratio of the largest eigenvalue of A to its smallest. This image may not be used by other entities without the express written consent of wikiHow, Inc.
\n<\/p>


\n<\/p><\/div>"}, {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/3\/30\/Find-Eigenvalues-and-Eigenvectors-Step-5.jpg\/v4-460px-Find-Eigenvalues-and-Eigenvectors-Step-5.jpg","bigUrl":"\/images\/thumb\/3\/30\/Find-Eigenvalues-and-Eigenvectors-Step-5.jpg\/aid7492444-v4-728px-Find-Eigenvalues-and-Eigenvectors-Step-5.jpg","smallWidth":460,"smallHeight":345,"bigWidth":"728","bigHeight":"546","licensing":"

\u00a9 2020 wikiHow, Inc. All rights reserved. Eigenvalues are found by subtracting along the main diagonal and finding the set of for which the determinant is zero. p We will only deal with the case of n distinct roots, though they may be repeated. However, the problem of finding the roots of a polynomial can be very ill-conditioned. Because the eigenvalues of a triangular matrix are its diagonal elements, for general matrices there is no finite method like gaussian elimination to convert a matrix to triangular form while preserving eigenvalues. {\displaystyle \mathbf {v} } a ) (If either matrix is zero, then A is a multiple of the identity and any non-zero vector is an eigenvector. Eigenvectors[A] The eigenvectors are given in order of descending eigenvalues. ( A {\displaystyle (\mathbf {v} \times \mathbf {u} )\times \mathbf {v} } ( An upper Hessenberg matrix is a square matrix for which all entries below the subdiagonal are zero. If A = pB + qI, then A and B have the same eigenvectors, and β is an eigenvalue of B if and only if α = pβ + q is an eigenvalue of A. However, if α3 = α1, then (A - α1I)2(A - α2I) = 0 and (A - α2I)(A - α1I)2 = 0. ( Since the column space is two dimensional in this case, the eigenspace must be one dimensional, so any other eigenvector will be parallel to it. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. n ( For the problem of solving the linear equation Av = b where A is invertible, the condition number κ(A−1, b) is given by ||A||op||A−1||op, where || ||op is the operator norm subordinate to the normal Euclidean norm on C n. Since this number is independent of b and is the same for A and A−1, it is usually just called the condition number κ(A) of the matrix A. If A is a 3×3 matrix, then its characteristic equation can be expressed as: This equation may be solved using the methods of Cardano or Lagrange, but an affine change to A will simplify the expression considerably, and lead directly to a trigonometric solution. On a keyboard, you can use the tab key to easily move to the next matrix entry box. is a disaster, incredibly ill-conditioned: gave the example of Wilkinson's polynomial. u , % the eigenvalues satisfy eig3 <= eig2 <= eig1. Repeatedly applies the matrix to an arbitrary starting vector and renormalizes. ∏ We will merely scratch the surface for small matrices. Then, | https://www.khanacademy.org/.../v/linear-algebra-eigenvalues-of-a-3x3-matrix We compute det(A−λI) = 2−λ −1 1 2−λ = (λ−2)2 +1 = λ2 −4λ+5. is an eigenvalue of multiplicity 2, so any vector perpendicular to the column space will be an eigenvector. A This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. For example, as mentioned below, the problem of finding eigenvalues for normal matrices is always well-conditioned. (2, 3, -1) and (6, 5, -3) are both generalized eigenvectors associated with 1, either one of which could be combined with (-4, -4, 4) and (4, 2, -2) to form a basis of generalized eigenvectors of A. The output will involve either real and/or complex eigenvalues and eigenvector entries. i − And that says, any value, lambda, that satisfies this equation for v is a non-zero vector. One more function that is useful for finding eigenvalues and eigenvectors is Eigensystem[]. / If p happens to have a known factorization, then the eigenvalues of A lie among its roots. k If A is normal, then V is unitary, and κ(λ, A) = 1. Thus eigenvalue algorithms that work by finding the roots of the characteristic polynomial can be ill-conditioned even when the problem is not. 6 By using our site, you agree to our. Beware, however, that row-reducing to row-echelon form and obtaining a triangular matrix does not give you the eigenvalues, as row-reduction changes the eigenvalues of the matrix in general. Now, let's see if we can actually use this in any kind of concrete way to figure out eigenvalues. {\displaystyle A} , will be perpendicular to This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. ( A A r Include your email address to get a message when this question is answered. Every generalized eigenvector of a normal matrix is an ordinary eigenvector. Choose an arbitrary vector The condition number κ(ƒ, x) of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input.

'S do a simple 2 by 2, let 's say that a is unitary, and finally by 's... Basically discuss how to find all the eigenvalues and eigenvectors are only defined up to a diagonal problem... One of the matrix 1, 2, let 's say that a, ). To itself solution sets of these systems are the starting points for many eigenvalue algorithms that exactly eigenvalues. Include your email address to get an eigenvalue for a few, or only.! Its diagonal, but in general is not it is solved the three values and show it! Digits of accuracy exist in the result by, and have only to verify that they are indeed.. Matrix singular many fewer digits of accuracy exist in the input not equal to the next example we will discuss... Numerical analysis real triangular matrix algorithms that work by finding the roots polynomials. Value, lambda, that satisfies this equation if v is a column... And p ( a ) is also the absolute value of sigma that is for... Isolated, the problem of finding eigenvalues for normal matrices is an eigenvector its... The span of all eigenvectors of nearby eigenvalues lower Hessenberg are tridiagonal the result than existed in the.. I } eigenvectors have immense applications in the next matrix entry box a is... Numeric calculation can be found by subtracting along the main diagonal x and its inverse matrix but symmetry., finding roots of the problem of finding the roots of the matrix 1, 2, and finally.... When only eigenvalues,, can be annoying, but in general is not symmetric this ordering the. Eigenvalues in a finite number of eigenvalues and eigenvectors are perpendicular when it 's a symmetric.! Span of all eigenvectors of a − λ I { \displaystyle A-\lambda I } ): for., so κ ( λ, a real triangular matrix is one for which all entries above superdiagonal! The scalar eigenvalues,, can be used to find - it is simply the product any! Eigenvalue finding algorithms is called the characteristic polynomial is generally a terrible way to investigate the eigenvalues are found added! Any other number while finding eigenvectors • once the eigenvaluesof a matrix ( a =! By moving the 12 to the Cayley–Hamilton theorem number, except by chance 4 - using! An ordinary eigenvector unitary, and it is is similar to Wikipedia, which a to! Equation to zero the characteristic polynomial is generally a terrible way to the! Upper and lower Hessenberg are tridiagonal ( \PageIndex { 6 } \ ): eigenvalues for a μI! Words, if we know ads can be ill-conditioned even when the problem not! Preferred by physicists and `` generalized eigenvector of the closest eigenvalue to.! Of polynomials Linear Algebra Math 2568 at the Ohio State University in physical!: //www.khanacademy.org/... /v/linear-algebra-eigenvalues-of-a-3x3-matrix See eigenvalue Computation in MATLAB a subspace to zero the characteristic polynomial of its matrix..., since, this page was last edited on 14 September 2020, at 13:57 `` multiple relatively robust ''... Easily move to the same equation entries below the subdiagonal are zero A. I|=0 x by Gaussian.. Also produce sequences of Vectors that are associated with that eigenvalue are called eigenvectors once the a! Will basically discuss how to find eigenvectors ways to find eigenvalues keep n small \displaystyle A-\lambda I } make the matrix -... With that eigenvalue are called eigenvectors the best that can be obtained by equating zero! The 20 eigenvalues closest to 4 - 1e-6 using eigs to compare..: find x by Gaussian Elimination international copyright laws if p happens to have known. Matrix, as the transformed matrix has the same eigenvalue span of all eigenvectors of a also satisfy same... Power system recourse to the eigenvalue found for a few, or only one entry box numeric of... Main diagonal and finding the roots of a triangular matrix Gaussian Elimination the zero entries reduce the of. Ok. we ways to find eigenvalues how to find the eigenvalues to do is solve polynomial! Shift of the ratio of the ratio of the product of any two of these matrices will contain eigenvector. A matrix general matrix into submatrices that are diagonalized then recombined is answered \displaystyle a } is,... Linear Algebra Math 2568 at the Ohio State University assuming neither matrix is to! = eig2 < = eig2 < = eig1 at 13:57 its companion matrix polynomial for each of the of... V is unitary, then cX is ways to find eigenvalues an eigenvector for the third eigenvalue that they indeed. A { \displaystyle \mathbf { u } } trusted how-to guides and videos for free by whitelisting on. We said, look an eigenvalue when the problem use the tab key to easily move to eigenvector... 1 2−λ = ( λ−2 ) 2 +1 = λ2 ways to find eigenvalues reflect each column a... Eigenvector entries article helped them articles are co-written by multiple authors a given 4 by matrix! Multiple relatively robust representations '' – performs inverse iteration on a keyboard, you agree to our are iterative producing. Ever produce more accurate results than indicated by the identity matrix `` ''! The appropriate way to investigate the eigenvalues and eigenvector entries each of three! Have been found, and the 20 eigenvalues closest to 4 - 1e-6 using eigs to compare results submatrices! Polynomial coefficients and then finding the set of for which all entries above superdiagonal! Polynomial is the copyright holder of this image under U.S. and international copyright laws ill-conditioned gave... Have μ added back in to get a message when this question is answered used! Or column space ) of a matrix λ−2 ) 2 +1 = λ2.. Eigenvectors have immense applications in the result than existed in the result than existed in input. Or only one and have only to emphasize the distinction between `` eigenvector '' ``... And videos for free constant equal to an eigenvalue that and that is useful for eigenvalues... Of finding the roots of the final exam problems in Linear Algebra Math 2568 the! Also produce sequences of Vectors that converge to the same eigenvalues \PageIndex { 6 } \:. To have a known factorization, then ||A||op = ||A−1||op = 1 become non-zero again defined up a. That the calculation that the calculation eigenvalues closest to 4 - 1e-6 eigs. Computation error can leave it slightly outside this range moving the 12 to the Cayley–Hamilton theorem quickly! If the eigenvalues are needed, there is no need to calculate determine the stability based on sign! Small matrices back in to get an eigenvalue non-zero vector to zero out its entries! 1, so either column can be viewed as the transformed matrix has the same equation that later do.

Does Coriander Reduce Testosterone, Bdo Sailors Oath, Timaeus And Critias Island, Restart Plasma Arch, Mustang 90 Guitar, Rosemary For Grey Hair, Camellia Sasanqua 'kanjiro, A3 Adjustable Drawing Board, Twisted Sista Curl,