"Eigenvalues tell you about powers of a matrix in a way that we had no way to approach previously."
"And now what do I do? Well, before I even think, I find its eigenvalues and eigenvectors."
"Do you see that we, we've got precise information about the Fibonacci numbers out of the eigenvalues?"
"That will tell -- those eigenvectors -- the eigenvalues will already tell you what's happening."
"The vector that doesn't change directions after linear transformation is an eigenvector of the matrix."
"Every zero eigenvalue corresponds to some kind of invariance in your system."
"This is where we show you how it relates to eigenvalues and eigenvectors."
"Every matrix is similar to its Jordan form."
"The eigenvalues of the Hecke operators are algebraic integers."
"The eigenvalues have an imaginary component, so it's going to be damped and it's going to be oscillating at that frequency."
"The eigenvectors are giving you the fundamental modes of vibration of that object."
"Lambda is equal to Lambda bar, and so Lambda belongs to the reals."
"When the matrix acts on this vector, the vector stays the same; it's just multiplied by a value."
"If you're Hermetian, your eigenvalues are all real."
"The eigenvalues of Hermitian operators are real numbers, which is important because physical quantities are typically described by real numbers."
"The eigenvalues of a symmetric matrix are real, and the eigenvectors are perpendicular."
"The spectrum is the set of eigenvalues of a matrix."
"The eigenvalues are telling me something important about the answer."
"Once we get small numbers here, little Epsilons, what would you expect to see on the diagonal? The eigenvalues."
"The determinant of A minus lambda I is equal to zero."
"If we want to maximize the amount of variance then we must choose the eigenvector corresponding to the largest eigenvalue of the matrix."
"If the radius of these eigenvalues of this \( M \) matrix are less than one, then this will be stable."
"An operator is diagonalizable if it has a set of eigenvectors that span the space."
"The eigenvalues of random matrices generally do not like to stay too close to each other; they talk to each other even though the entries are independent."
"If my Delta X is small enough, I can bound a region around each of these eigenvalues where my new eigenvalues in this nonlinear system will be."
"When we measure Q in a system with a discrete quantum mechanical spectrum, we always get one of the eigenvalues of the operator corresponding to the observable that we measure."
"We can solve this system in closed form by finding the eigenvalues and eigenvectors of this matrix."
"The definiteness of a given matrix is governed by the sign of all of the eigenvalues."
"Only positive eigenvalues, you have a positive definite matrix."
"If you have eigenvalues which are greater than or equal to zero, you have a positive semi-definite matrix."
"If all the eigenvalues are positive, or positive or greater than equal to zero, that's the same then x star as a local minimum."
"Has all eigenvalues strictly greater than zero, then you have a strict local minima."
"Principal directions are the eigenvectors of the shape operator; the principal curvatures are the eigenvalues."
"Eigenvalues and eigenvectors are not mystical; they are specifically designed to do this."
"We sort the eigenvectors according to their eigenvalues in decreasing order."
"Eigenvalues and eigenvectors are not exclusive to operators in state space; they are useful in many different areas."
"Any instability, any positive eigenvalues will always dominate and win."
"The eigenvalues of this A matrix determine everything about the solution to this differential equation."
"Eigenvalues tell us how these systems change in time, what grows, what decays, how fast, and in what direction."
"The eigenvalues of MA are very closely connected with the order of A."
"By using phase estimation to approximate these eigenvalues closely enough we can determine the order of A."
"If the vector doesn't change length, then its eigenvalue will be 1."
"The trajectories leave along the direction corresponding to the eigenvalue closest to zero."
"And lo and behold, the eigenvalues are one, two, three, four."
"When the going gets tough, the eigenvalues and the eigenvectors just get going."
"The determinant of a matrix is essentially a product of eigenvalues."
"If you're a scientist or engineer, you've definitely come across eigenvalue problems before."
"That's the great fact about diagonalizing, that's how the eigenvectors pay off."
"This is awesome, so we now have this family of eigenvalues and a family of eigenfunctions."
"If all the eigenvalues Lambda of a are real and distinct, then essentially my eigenvectors T span RN."
"We've seen two different types of A matrices with repeated eigen values, and they have very different behavior."
"The only possible eigenvalues for an orthogonal transformation are plus or minus one."
"The roots of this polynomial, by definition, are the eigenvalues of a matrix A."
"The best \(X\) is the eigenvector, and the ratio is the eigenvalue."
"The eigenvalues in the matrix case for \( Kx = \lambda Mx \), the eigenvalue problem, lambda the lowest eigenvalue, has a nice property."
"That's a good example, and then is the matrix positive definite? The eigenvalues are sitting there, two and five, both positive."
"These are the good matrices, symmetric matrices, their eigenvalues lie on the real line."
"Everything you know about a matrix shows up somehow in its eigen vectors and eigen values."
"In first order perturbation theory, the eigenvalue is simply the expectation value of the perturbation over the wave functions you get from the unperturbed operator."
"When we measure an observable, the only possible outcome is one of the eigenvalues, with probability equal to the absolute value of the expansion coefficients squared."
"The energy eigenvalue equation reduces to a simple 1-D effective energy eigenvalue equation with a simple dr squared, and an effective potential."
"Assume \( A \) is diagonalizable, we have \( T^{-1}AT = \Lambda \)."
"The characteristic polynomial in terms of the eigenvalues is just this thing worked out here."
"The maximum value on the unit sphere will be the greatest eigen value lambda 1."
"The minimum value on the unit sphere will be the least eigen value lambda n of a."
"The largest coefficients and the diagonal matrix turn out to be the same thing as the largest eigen values."
"If all of my eigenvalues have negative real part, then all of my dynamics are stable."
"The stability of your system in continuous-time or discrete-time completely depends on the eigenvalues of your matrix."
"The Harris corner detector... calculates the two-dimensional plot at every location in the image and calculates eigenvalues, and if you have two large eigenvalues, you have a corner."
"The spectrum of a matrix and the spectrum of its transpose are the same."
"The singular value decomposition can be seen as a generalization of the notion of eigenvalues and eigenvectors."
"We would be determining a scalar lambda and a non-zero vector x such that this matrix equation AX is equal to lambda X is satisfied."
"If S squared equals S, then the only possible eigenvalues of that matrix S are zero and one."