Determinant Definition and 494 Threads

I am really really really confused how to read and construct Slater determinants :( Can someone please explain it using He at the ground state (1s2) and He at excited state (1s12s1) ?

Hey all, the determinant is used in interseting places and I know how to use it but I don't know the intuition of it,Can anyone elucidate it for me? Any help apreciated.

EDIT: I figured out what I was doing wrong, trivial mistake. If this could get deleted that would be good. so using {x,y,z} I'm making a vandermonde matrix; https://en.wikipedia.org/wiki/Vandermonde_matrix. When calculating the determinant by cofactor expansion I calculate the determinant to be...

I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...

I am attempting to find the determinant using gaussian elimination for the following matrix [4 3 2; 1 7 8; 3 9 3]. I have begun by attempting to form zeros below the diagonal. The answer should be -165, however i keep getting values up to -665. I have tried a variety of times and keep getting...

As a step in a solution to another question our lecture notes claim that the matrix (a,b,c,d are real scalars). \begin{bmatrix} 2a & b(1+d) \\ b(1+d)& 2dc \\ \end{bmatrix} Is positive definite if the determinant is positive. Why? Since the matrix is symmetric it's positive definite if the it...

I attempting to find the determinant using gaussian elimination for the following matrix [1 2 3; 3 2 2; 0 9 8]. I have begun by attempting to form zeros below the diagonal. My first row operation was to make row 2 equal to 3(row 1) - row 2. This gives me [1 2 3; 0 4 7; 0 9 8] . I think i am...

Homework Statement If A is a square matrix of order 3 then the true statement is 1. det(-A) = - det A 2.det A = 0 3.det ( A + I) = I + detA 4.det(2A) = 2detA Homework Equations NA The Attempt at a Solution 2. option is obviously not true. Making a random matrix A and verifying properties 1. ...

I was able to prove a), but I am unsure how to prove b. Is there some sort of geometric interpretation I should be aware of?

I understand the intuition behind it, but I'm unable to prove it. Essentially, three non-colinear points define a plane, and so by adding one more point on the plane, it becomes dependent. This means that the determinant is 0, since there is probably a dependent row lying around. Also, how is...

Homework Statement Please see the attached file if my inline insertion does not work. Homework Equations ##det(A)=det(A^T)##[/B]The Attempt at a Solution Since a matrix has a determinant of zero only when it's columns are linearly dependent, we look for a set of points [x1 x2] such that...

We have a system of 2 particles, let's say with following hamiltonian: $$\hat{H} = -\frac{\hbar^2\hat{\nabla}_1^2}{2m} -\frac{\hbar^2\hat{\nabla}_2^2}{2m} $$ The eigenstates are often represented as (spatial wavefunction)*(spin wavefunction), where the spin wavefunction is a singlet or a...

Hi, the question (from math methods for physicists) is: If A is orthogonal and det(A)=+1, show that (det A)aij = Cij(A). I know that if det(A)=+1, then we are looking at a rotation. (Side question - I have seen that det(A) =-1 can be a reflection, but is 'mostly not reflections'; what does...

I don't quite follow this, can anyone explain?

I am trying to find the determinant of the following via upper triangular form: $$\left[\begin{array}{c}-1 & -1 & 1 & 0 \\ 2 & 1 & 1 & 3 \\ 0 & 1 & 1 & 2 \\ 1 & 3 & -1 & 2 \end{array}\right]$$Using row reduction to bring it to upper triangular matrix: $$\left[\begin{array}{c}-1 & -1 & 1 & 0 \\...

How do you represent einstein field equations with levi civita symbol or jacobian determinant? I saw a lot of work that involves this but I don't know how and why. Besides how is the jacobian determinant related to the levI civita symbol?

Hi, How would you solve a singular matrix? ie when determinant is zero. Lets assume an equation, (1) Ax+By=E and (2) Cx+Dy=F if the determinant; AD-BC = 0, and therefore the matrix is singular, How to go around solving the equation? LU decomposition, Gaussian elimination? Ideally I am...

I have attached the properties/rules that will aid in simplifying the 3 x 3 determinant so that it can be calculated with minimum effort. I am trying to understand rule (2) and (4). I am not so clear about the idea of linearity. Can anyone please explain these two rules with regards to example...

For a 2x2 matrix, does the general definition hold? If so, how exactly is the minor ## M_{i,j} ## computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?

If I switch 2 rows, do I have to multiple by -1 each time? For example, I have If I switch row 2 and 3, will it become this: Or this? Each time I make a switch, do I have to also put a negative sign? Edit: Not really related to Gaussian elimination, but this is from a Gaussian...

Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...

Any1 can explain in an easy way?I know is something to do with the facts that row operations do not change the determinant of a matrix and also determinant of A=determinant of A transpose.But just dk how to prove in full sentence. Appreciate any1 help.

Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t)Homework Equations W(y1, y2) = |y1 y2...

the first row 1 0 0 2 the 2nd row 0 1 2 0 the 3rd row 0 2 1 0 the 4th row 2 0 0 1 I would like to ask which is the most efficient way of solving this ques.Though i can solve but is long method, I know there must have some quick 1, appreciate if u can share it. thank you

Is this just a coincidence that cross product can be found from determinant of 3*3 matrix? what is the differences between wedge product and cross product?Thanks.

Hello, If we let U and V be two single determinant wave functions built up of spin orbitlas ui and vj respectively, will the overlap between them be as follows: <U|V> = Det{<ui|vi>} Thank you

Homework Statement Let R be the ring of all 2*2 matrices over Zp , a prime. Show that for x, y contained in R, det(xy) = det(x) det(y). Homework Equations The Attempt at a Solution The det(xy)≠0, therefore the equality can be true. However, I am not sure how to prove that the...

[SIZE="4"]Definition/Summary A Slater determinant is a representation of a many-particle wave function for a system of fermions, which satisfies the anti-symmetry requirement. In other words, that the wave function changes sign on interchange of two particle coordinates (e.g...

Under a change of variables: x^{\mu} \rightarrow x^{\mu}+ \delta x^{\mu} How can I see how the determinant of the metric changes? \sqrt{|g(x)|}? Is it correct to see it as a function? f(x) \rightarrow f(x+ \delta x) = f(x) + \delta x^{\mu} \partial_{\mu} f(x) ?

It seems to me that if a row is able to be zeroed out through Gaussian reduction that the determinate of that matrix would equal zero. Therefore, we know that at least one of equations/vectors that constructed the matrix was formed from the other two rows. That is -- that equation is dependent...

Browsing in the wiki, I found those formulas: http://en.wikipedia.org/wiki/Determinant#Relation_to_eigenvalues_and_trace So, my doubt is: if is possible to express the determinant in terms of the trace, thus is possible to express the trace in terms of the determinant too?

Given the following: $$\\ \begin{bmatrix} A & 0\\ 0 & B\\ \end{bmatrix}$$ the eigenvalues is exactaly A and B. So analogously, is possible to write a matrix with only two elements, T and D, such that the trace is T and the determinant is D? I tried something: $$\\ \text{tr} \left(...

Hi there! How can I prove that the space of matrices (2x2) nonzero determinant is dense in the space of matrices (2x2) ? I've already proved that it's an open set. Thanks. PD: Sorry about the mistake in the title.

Homework Statement For a system Ax= 0, suppose det (A)= .0001. Which of the following describes the solutions to system? There is exactly one solution, but the system is close to having infinity many. There is exactly one solution, but the system is close to having none. Homework Equations...

Hi. I am reading a physics text, and in one of the lines it uses the following relation: \mathrm{det}(\delta^\mu_\lambda +\frac{\partial \delta x^\mu}{\partial x^\lambda}) = 1 + \mathrm{Tr}\frac{\partial \delta x^\mu}{\partial x^\lambda} where \mu and \lambda are matrix elements, and...

Homework Statement Theorem. Let A be a k by k matrix, let D have size n by n and let C have size n by k. Then det \left(\begin{array}{cc}A&0\\C&D\end{array}\right) = (det A)\cdot (det D). Proof. First show that \left(\begin{array}{cc}A&0\\0&I_{n}\end{array}\right) \cdot...

Homework Statement Determinant |a b c| |d e f| = 5 |g h i| What is the determinant of? |4a 4b 4c| |3d 3e 3f| | g h i| The Attempt at a Solution So far I got this, a(ei - hf) - b(di - gf) + c(dh - ge) = 5 How would I even go about this, I have 9 unknowns, but only 1 equations?

I'm doing a calculation which finds the characteristic polynomial of a matrix, HH, with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 it finishes evaluating the Discriminant command within a few hours, but at a 10x10, which...

Hi, In his original paper, Schwarzschild set the "'equation of the determinant" to be: |g|=-1. In other words, he imposed the determinant of the metric to be equal to minus one when solving the Einstein's equations. Must we impose this equality systematically in general relativity and why...

I need to prove that a 3x3 matrix with all odd entries will have a determinant that is a multiple of 4. This is how I set it up: I let A = { {a, b, c}, {d, e, f}, {g, h, i} } with all odd entries then I define B = { {a, b, c}, {d + na, e + nb, f + nc}, {g + ma, h + bm, i + cm} } where I add...

If all columns of a matrix are unit vectors, the determinant of the matrix is less or equal 1 I am trying to prove this assertion,which i guess to be true. can anybody help me? Thank's in advance

Homework Statement I need to prove or show that this n-rowed determinant which corresponds to cosine multiple angle formula is in fact true using induction. The Attempt at a Solution First let ##a = \cos \theta## and suppose I have this n by n determinant. $$ \begin{vmatrix} &a &1 &0& \\...

Homework Statement Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint) show that det AB = detAdetB Homework Equations The Attempt at a Solution Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...

Homework Statement find the minima and maxima of the following function: ##f:\mathbb{R}^3 \to \mathbb{R} : f(x,y,z)=x(z^2+y^2)-yx## The Attempt at a Solution after computing the partials, i see ∇f=0 for every point in the x-axis: (a, 0, 0) The Hessian is: ( 0 0 0 ) ( 0 2a -1...

Homework Statement Show by direct expansion that: det (I + εA) = 1 + εTr(A) + O(ε2) Homework Equations f(x) = f(a) + (x-a)f'(a) + (1/2)(x-a)2f''(a) + ... The Attempt at a Solution Does the question mean Taylor expansion when they say 'direct expansion'? I'm kind of stuck on...

Homework Statement Entries of a 2X2 determinant are chosen from the set {-1,1}. The probability that determinant has zero value is The Attempt at a Solution Zero value can occur if all the elements of first row is 0. The remaining 2 places can be filled in 3*3=9 ways. Now other...

Solve for x x -6 -1 2 -3x x-3 = 0 -3 2x x+2Attempt- x-2 3(x-2) -(x-2) 2 -3x x-3 -3 2x x+2 1 3 -1 2 -3x x-3 -3 2x x+2 1 3 -1 0 -3x-6 x-1 = 0 0 -x+3 2(x-1) x =-3 By doing this I'm getting just one value of x. How do i get the values of x like 2 and 1?

I know that 3 x 3 determinant gives the volume of a parallelopiped, but how come after the row operations also it's gives the Same volume when it's elements are changed or in another words it's sides are being modified?

Hi, I have a question about describing geometrically the action of an arbitrary orthogonal 3x3 matrix with determinant -1. I would like to know if my proposed solutions are satisfactory, or if they lack justification. I have two alternate solutions, but have little confidence in their validity...

1. Consider the 2x2 matrix \sigma^{\mu}=(1,\sigma_{i}) where \sigma^{\mu}=(1,\sigma) where 1 is the identity matrix and \sigma_{i} the pauli matrices. Show with a direct calcuation that detX=x^{\mu}x_{\mu} 3. I'm not sure how to attempt this at all...