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Do similar matrices have the same eigenvalues?
Yes, similar matrices have the same eigenvalues.
What is the condition for the addition of matrices?
The matrices must have the same dimensions.
Can matrices of the same dimension be multiplied?
No. The number of columns of the first matrix needs to be the same as the number of rows of the second.So, matrices can only be multiplied is their dimensions are k*l and l*m. If the matrices are of the same dimension then the number of rows are the same so that k = l, and the number of columns are the same so that l = m. And therefore both matrices are l*l square matrices.
What is linear combination in matrices?
If X1, X2 , ... , Xn are matrices of the same dimensions and a1, a2, ... an are constants, then Y = a1*X1 + a2*X2 + ... + an,*Xn is a linear combination of the X matrices.
Can a Hermitian Matrix possess Complex Eigenvectors?
Yes. Simple example: a=(1 i) (-i 1) The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1). A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
Do similar matrices have the same eigenvalues?
Yes, similar matrices have the same eigenvalues.
How can you prove that similar matrices have the same trace?
you tell me
Do commutative matrices have the same eigenvectors?
It is true that diagonalizable matrices A and B commute if and only if they are simultaneously diagonalizable. This result can be found in standard texts (e.g. Horn and Johnson, Matrix Analysis, 1999, Theorem 1.3.12.) One direction of the if and only if proof is straightforward, but the other direction is more technical: If A and B are diagonalizable matrices of the same order, and have the same eigenvectors, then, without loss of generality, we can write their diagonalizations as A = VDV-1 and B = VLV-1, where V is the matrix composed of the basis eigenvectors of A and B, and D and L are diagonal matrices with the corresponding eigenvalues of A and B as their diagonal elements. Since diagonal matrices commute, DL = LD. So, AB = VDV-1VLV-1 = VDLV-1 = VLDV-1 = VLV-1VDV-1 = BA. The reverse is harder to prove, but one online proof is given below as a related link. The proof in Horn and Johnson is clear and concise. Consider the particular case that B is the identity, I. If A = VDV-1 is a diagonalization of A, then I = VIV-1 is a diagonalization of I; i.e., A and I have the same eigenvectors.
What is the condition for the addition of matrices?
The matrices must have the same dimensions.
Can matrices of the same dimension be multiplied?
No. The number of columns of the first matrix needs to be the same as the number of rows of the second.So, matrices can only be multiplied is their dimensions are k*l and l*m. If the matrices are of the same dimension then the number of rows are the same so that k = l, and the number of columns are the same so that l = m. And therefore both matrices are l*l square matrices.
How can I calculate eigenvectors in MATLAB?
To calculate eigenvectors in MATLAB, you can use the "eig" function. This function returns both the eigenvalues and eigenvectors of a given matrix. Simply input your matrix as an argument to the "eig" function, and it will output the eigenvectors corresponding to the eigenvalues.
How can Mathematica be used to compute and normalize eigenvectors of a given matrix?
Mathematica can be used to compute and normalize eigenvectors of a given matrix by using the Eigensystem function to find the eigenvectors and eigenvalues of the matrix. Then, the Normalize function can be applied to normalize the eigenvectors.
How can I use the numpy diagonalize function to find the eigenvalues and eigenvectors of a matrix in Python?
To find the eigenvalues and eigenvectors of a matrix using the numpy diagonalize function in Python, you can first create a matrix using numpy arrays. Then, use the numpy.linalg.eig function to compute the eigenvalues and eigenvectors. Here's an example code snippet: python import numpy as np Create a matrix A np.array(1, 2, 3, 4) Compute eigenvalues and eigenvectors eigenvalues, eigenvectors np.linalg.eig(A) print("Eigenvalues:", eigenvalues) print("Eigenvectors:", eigenvectors) This code will output the eigenvalues and eigenvectors of the matrix A.
What is the syntax for calculating eigenvalues and eigenvectors in MATLAB in a specific order using the 'eig' function?
To calculate eigenvalues and eigenvectors in MATLAB using the 'eig' function, the syntax is as follows: eigenvectors, eigenvalues eig(matrix) This command will return the eigenvectors and eigenvalues of the input matrix in a specific order.
What is linear combination in matrices?
If X1, X2 , ... , Xn are matrices of the same dimensions and a1, a2, ... an are constants, then Y = a1*X1 + a2*X2 + ... + an,*Xn is a linear combination of the X matrices.
Can a Hermitian Matrix possess Complex Eigenvectors?
Yes. Simple example: a=(1 i) (-i 1) The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1). A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
What has the author Jan R Magnus written?
Jan R. Magnus has written: 'Linear structures' -- subject(s): Matrices 'The bias of forecasts from a first-order autoregression' 'The exact multiperiod mean-square forecast error for the first-order autoregressive model with an intercept' 'On differentiating Eigenvalues and Eigenvectors' 'The exact moments of a ratio of quadratic forms in normal variables' 'Symmetry, 0-1 matrices, and Jacobians'
