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Using MATLAB to find matrix eigenvalue and eigenvector A=[1 5 1 3 3 1/5 1 6 4 2 1 1/6 1 3 4 1/3 1/4 1/3 1 2 1/3 1/2 1/4 1/2 1 ]
Input: x = [1 5 1 3 3; 1 / 5 1 6 4 2; 1 1 / 6 1 3 4; 1 / 3 1 / 4 1 / 3 1 2; 1 / 3 1 / 2 1 / 4 1 / 2 1] eig (x) output: ans = 6.3156 - 0.5309 + 2.7527i - 0.5309 - 2.7527i - 0.1269 + 0.4050i - 0.1269 - 0.4050i
Using MATLAB to find the eigenvector corresponding to the maximum eigenvalue of matrix
Matrix: a = [1 2 3 4; 1 / 2 1 2 3; 1 / 3 1 / 2 1 2; 1 / 4 1 / 3 1 / 2 1], find the eigenvector corresponding to the maximum eigenvalue of the matrix!
>> [d,v]=eig(A)
d =
-0.8135 -0.8493 -0.8493 -0.7038
-0.4826 0.0004 - 0.4268i 0.0004 + 0.4268i 0.5934
-0.2787 0.2498 - 0.0499i 0.2498 + 0.0499i -0.3592
-0.1661 0.0252 + 0.1758i 0.0252 - 0.1758i 0.1535
v =
4.0310 0 0 0
0 -0.0017 + 0.3533i 0 0
0 0 -0.0017 - 0.3533i 0
0 0 0 -0.0276
The maximum characteristic value is 4.0310
The first column of V is the corresponding eigenvector
Finding eigenvalues and eigenvectors of matrix (0 a 0 a)
┏0 a┓
Γ 0 a Γ is an upper triangular matrix with eigenvalues: λ 1 = 0, λ 2 = a
① , a = 0, λ 1 = λ 2 = 0, eigenvector: any (x, y) ′≠ 0
② A ≠ 0, λ 1 = 0, eigenvector: (k, 0) ′ (K ≠ 0)
λ 2 = a, eigenvector: (k, K) ′ (K ≠ 0)
Let a and B be 3-dimensional column vectors and satisfy the transpose * b = 5 of a, then what is the eigenvalue of transpose * a of matrix B
Eigenvalues of matrix b'a
B'a is not a matrix, so it should not be so simple. I guess ba'a is still 5 because ba'b = 5B = > its eigenvalue (because if the eigenvector is u and the eigenvalue is V, then ba'u = vu, but a'u
Let a = (α, 2, γ 2, 3, γ 3), B = (β, γ 2, γ 3), where α, β, γ 2, γ 3 are all three-dimensional column vectors, and / A / = 18. / B / = 2. Find / a-b/
The answer is 2, using the determinant property as shown in the figure. The economic mathematics team will help you solve it, please adopt it in time
On the problem of Linear Algebra: let α and β be 3-dimensional column vectors, and β ^ t be the transpose matrix of β. Then α * β ^ t is a matrix of order n with rank 1. Why?
I'm here to give you a proof: R (α * β ^ t) can be regarded as two matrices to multiply. A three-dimensional column vector is actually a matrix with three rows in one column. Then there is the formula R (AB) ≤ min (R (a), R (b)). However, R (a) r (b) is a non-zero one-dimensional column vector, and its rank is 1, and R (α * β ^ t) this
Let α 1 and α 2 be eigenvectors of matrix A belonging to different eigenvalues. It is proved that α 1 + α 2 are not eigenvectors of matrix A
It is proved that let α 1 and α 2 be the eigenvectors of a with different eigenvalues λ 1 and λ 2
Then a α 1 = λ 1 α 1, a α 2 = λ 2 α 2, and λ 1 ≠ λ 2
Suppose α 1 + α 2 are the eigenvectors of a belonging to the eigenvector λ
Then a (α 1 + α 2) = λ (α 1 + α 2)
So λ 1 α 1 + λ 2 α 2 = λ (α 1 + α 2)
So (λ - λ 1) α 1 + (λ - λ 2) α 2 = 0
Because the eigenvectors of a belong to different eigenvalues are linearly independent
So λ - λ 1 = 0, λ - λ 2 = 0
Therefore, λ = λ 1 = λ 2 is contradictory to λ 1 ≠ λ 2
So α 1 + α 2 is not the eigenvector of A
Teacher, how can I prove that for each eigenvalue, the linearly independent eigenvectors that a matrix can have will not exceed the multiplicity of this eigenvalue
This is more troublesome
Eigenvalue problems of symmetric positive definite matrices 4
Recently, I'm a little ahead of myself in mathematics. I can't think of some problems clearly
Now I know that a symmetric positive definite matrix of order n must have n positive eigenvalues
3. For a positive definite matrix A, can it have n nonnegative eigenvalues?
(positive definite matrices don't require symmetry, am I right?)
A lot of questions are asked, mainly to know when there will be a series of orthogonal eigenvectors to represent the whole space
For an asymmetric matrix A, its eigenvalues may appear imaginary numbers, but no matter how it is, there are always imaginary numbers
μ_ min
Is it necessary and sufficient for this matrix to be positive definite if all eigenvalues are positive?
Such as the title
Yes, it's necessary and sufficient. In addition, the order of the bamboo formula is greater than 0. The original formula y = XT a x > 0 is necessary and sufficient
RELATED INFORMATIONS
- 1. A. It is proved that the eigenvalues of AB are all ≥ 0
- 2. Is the eigenvectors of any matrix with different eigenvalues necessarily orthogonal
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- 4. (number three) what are the rules of eigenvalues of symmetric matrices and how to find them? There is an example in Li Yongle's book, which says that a is a second-order matrix, and the four elements are all 1. Because a is a symmetric matrix, the eigenvalues of a are 2 and 0. How does he know the eigenvalues are 2 and 0? Is there any rule? It should not be derived from solving the eigenvalue equation? .
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- 7. A={x||x-1|
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- 10. Given the set M = {(x, y) | 4x + y = 6}, P = {(x, y) | 3x + 2Y = 7}, then m ∩ P equals () A. （1，2）B. {1}∪{2}C. {1，2}D. {（1，2）}
- 11. It is proved that the eigenvalues of real antisymmetric matrices are zero or pure imaginary numbers It doesn't matter if it's a bit wordy. I must understand it
- 12. A is a square matrix of order n. if there is a square matrix of order n, B is not equal to 0, so that ab = 0, it is proved that the rank of a is less than n
- 13. rt. It is proved that if matrix A is commutative with all n-order matrices, then a must be a quantity matrix, that is, a = AE
- 14. Let a and B be symmetric matrices of the same order, and prove that AB is symmetric if and only if a and B are commutative
- 15. It is proved that the orthogonal matrix whose eigenvalues are all real numbers is a symmetric matrix Mayday: how to prove it. I'm almost broken Don't talk nonsense. This question is the subject of postgraduate entrance examination for Advanced Algebra over the years. It must be correct. Otherwise, counter examples will be given
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- 17. It is proved that the function f (x) = X-1 / X decreases monotonically on (1, positive infinity)
- 18. Given a ∈ R, function f (x) = x | x-a |, problem (1) when a = 2, write the monotone increasing interval of function y = f (x) (2) find the maximum value of the function y = f (x) in the interval [0,2]
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- 20. If the monotone decreasing intervals of the function f (x) = (x-a) / x2 are (- infinity, 0) and (2, + infinity), then the real number a=________ .