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What does matrix commutativity mean?
A square matrix satisfying the commutative law of multiplication is called commutative matrix, that is, matrices A and B satisfy: a · B = B · a
What do two matrices represent interchangeably?
The square matrix satisfying the commutative law of multiplication is called commutative matrix, that is, the matrices A and B satisfy: a · B = B · a
Linear algebra (similar matrix)
Let a ∽ B, B have eigenvalues of 1, - 2, - 3. ① find the eigenvalue of a - & sup1;; ② find the eigenvalue of a adjoint
Are the eigenvalues of similar matrices the same
Is the eigenvalue of the inverse matrix the reciprocal of the original matrix
If we multiply the inverse by | a | and | a | = 1 × - 2 × - 3 = 6, the eigenvalue is 6 times of the inverse
Matrix A = 400 031 013 find an invertible matrix P such that P ^ - 1AP = Λis a diagonal matrix
Matrix A = 400
031
013 find an invertible matrix P such that P ^ - 1AP = Λis a diagonal matrix
Let the eigenvalue of the matrix a be λ, then | a - λ e | = 4 - λ 0003 - λ 1013 - λ is expanded according to the first line = (4 - λ) * (λ ^ 2-6 λ + 8) = 0, and the solution is λ = 2,4,4. When λ = 2, a-2e = 2 001 1011, the first line is divided by 2, and the second line ~ 1 001 1000 is subtracted from the third line to get the eigenvector (0,1, - 1
Let a be 460, negative 3, negative 50, negative 3, negative 61. Can a be diagonalized? If a can be diagonalized, find its invertible matrix P such that P is negative 1AP diagonal matrix
How can I ask again? The answer last time is not good. I am responsible for the end
First, the eigenvalues of a are obtained: - 2,1,1
Then the eigenvector corresponding to the eigenvalue is obtained
P = [-1 -2 0; 1 1 0; 1 0 1]
P^(-1)AP = diag{ -2,1,1}
The inverse of P = [1 20; - 1 - 10; - 1 - 21]
Calculate the results of the following matrices
λ 1 0
0 0 λ
This is Jordan's standard type. There is a standard algorithm
The n-th power is as follows:
λ^n n*λ^(n-1) 0
0 λ^n n*λ^(n-1)
0 0 λ^n
The requirement of matrix with square as unit matrix
N order
The requirement is to find all these matrices Hateful homework
It's very simple
A ^ 2-I = 0 means that the eigenvalue of a must be 1 or - 1, and a can be diagonalized, that is to say
A = P * diag{I_ k, -I_ {n-k}} * P^{-1},
Where k is an integer between 0 and n
Conversely, if a has the above form, then a ^ 2 = I
Let a satisfy the square of a = e, and prove that a + 2E is an invertible matrix
Since (a + 2e) (a-2e) = a ^ 2-4e = - 3E, (a + 2e) (- A / 3 + 2E / 3) = e, a + 2E is reversible
Let a and B be matrices of order n, e be identity matrices of order n, and ab = a-b. it is proved that a + e is invertible and ab = ba
AB+B=A
(A+E)B=A+E-E
(A+E)-(A+E)B=E
(A+E)(E-B)=E
So a + e is an invertible matrix
(A+E)(E-B)=(E-B)(A+E)=E
A-AB+E-B=A+E-BA-B
AB=BA
Is the sum of identity matrix and antisymmetric matrix invertible?
Please give detailed proof
Invertible, because the eigenvalue of antisymmetric matrix is 0 or pure imaginary number, so the eigenvalue of the sum of antisymmetric matrix and identity matrix is 1 or pure imaginary number, because the product of eigenvalues is not zero, so it is invertible
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