The eigenvalue spacing of iid random matrices
... Theorem (Sankar-Spielman-Teng, Tao-Vu, Rudelson-Vershynin) Let ξ be a normalized random variable and let N be an n × n random matrix with each entry an iid copy of ξ. Let M be a deterministic shift matrix. Then: P (sn (N + M) ≤ t) = O(n1/2 t) + error These results with a bound of t instead of t 2 ar ...
... Theorem (Sankar-Spielman-Teng, Tao-Vu, Rudelson-Vershynin) Let ξ be a normalized random variable and let N be an n × n random matrix with each entry an iid copy of ξ. Let M be a deterministic shift matrix. Then: P (sn (N + M) ≤ t) = O(n1/2 t) + error These results with a bound of t instead of t 2 ar ...
Systems of First Order Linear Differential Equations x1′ = a11 x1 +
... A system of linear (algebraic) equations, Ax = b, could have zero, exactly one, or infinitely many solutions. (Recall that each linear equation has a line as its graph. A solution of a linear system is a common intersection point of all the equations’ graphs − and there are only 3 ways a set of line ...
... A system of linear (algebraic) equations, Ax = b, could have zero, exactly one, or infinitely many solutions. (Recall that each linear equation has a line as its graph. A solution of a linear system is a common intersection point of all the equations’ graphs − and there are only 3 ways a set of line ...
The Fundamental Theorem of Linear Algebra
... Figure 6: There is no solution to Ax = b because b 6∈ C(A), but the projection p ∈ C(A) and there’s a solution to Ax̂ = p. The projection error e ∈ N (AT ) and N (A) is empty: it contains only 0. Figure 5: The projection error e is smallest when it’s perpendicular to C(A). is not perpendicular to C( ...
... Figure 6: There is no solution to Ax = b because b 6∈ C(A), but the projection p ∈ C(A) and there’s a solution to Ax̂ = p. The projection error e ∈ N (AT ) and N (A) is empty: it contains only 0. Figure 5: The projection error e is smallest when it’s perpendicular to C(A). is not perpendicular to C( ...
CHARACTERISTIC ROOTS AND FIELD OF VALUES OF A MATRIX
... Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of values of the matrix [14]. In an address delivered before the Mathematical Association of America in 1938, Browne [10] gave a su ...
... Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of values of the matrix [14]. In an address delivered before the Mathematical Association of America in 1938, Browne [10] gave a su ...
Solutions to HW 5
... Proof. We first prove the “only if” implication. So assume that T : V → W is an isomorphism; we first claim that then T(β) must be a linearly independent set of n vectors in W . To that end, write β = {v1 , . . . , vn }. Because T is one-to-one, the vectors T(vk ) are distinct for 1 ≤ k ≤ n, and thu ...
... Proof. We first prove the “only if” implication. So assume that T : V → W is an isomorphism; we first claim that then T(β) must be a linearly independent set of n vectors in W . To that end, write β = {v1 , . . . , vn }. Because T is one-to-one, the vectors T(vk ) are distinct for 1 ≤ k ≤ n, and thu ...
Lecture Notes - Computer Science at RPI
... vector x that minimizes Taking the derivative (I don’t necessarily expect that you can do this, but it isn’t hard) with respect to x, setting the result to 0 and solving implies Computing the SVD of A (assuming it is fullrank) results in Image Registration ...
... vector x that minimizes Taking the derivative (I don’t necessarily expect that you can do this, but it isn’t hard) with respect to x, setting the result to 0 and solving implies Computing the SVD of A (assuming it is fullrank) results in Image Registration ...
Home Work 3
... Minimal of the break up for all p is the optimal cost function C(I, I+k) that is being calculated by the algorithm. Hence the algorithm m correctly calculates C(1, n). 3. Some dietary fluid-packs have the following properties: ...
... Minimal of the break up for all p is the optimal cost function C(I, I+k) that is being calculated by the algorithm. Hence the algorithm m correctly calculates C(1, n). 3. Some dietary fluid-packs have the following properties: ...