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Deimantas
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Homework Statement
Diagonalize matrix
Homework Equations
The Attempt at a Solution
After diagonalization I get a diagonal matrix that looks like this
andrewkirk said:One way to tell is to build up the matrices A and B that represent the transformations that you preform in the diagonalisation process. If you've done that then you just need to perform the matrix multiplication ADB where D is the diagonal matrix, and check that it's equal to the original matrix M.
If the diagonal matrix is of eigenvalues (I can't recall whether they will be for general diagonalisation), another way might be to check that the characteristic equation of M is ##(\lambda-1)^2(\lambda-(x^5+x^4-1))##.
Deimantas said:Homework Statement
Diagonalize matrixView attachment 92744 using only row/column switching; multiplying row/column by a scalar; adding a row/column, multiplied by some polynomial, to another row/column.
Homework Equations
The Attempt at a Solution
After diagonalization I get a diagonal matrix that looks like this View attachment 92745 . What's the easiest way to tell if the answer is correct/incorrect?
Diagonalization of a matrix is a process through which a square matrix is transformed into a diagonal matrix. This transformation is achieved by finding a set of basis vectors that can be used to represent the matrix in a simpler form, with zeros in all off-diagonal elements.
Diagonalization of a matrix is important because it simplifies calculations involving the matrix, making it easier to solve equations and perform other operations. It also provides insight into the properties of the matrix, such as its eigenvalues and eigenvectors, which can be useful in various applications.
Diagonalization of a matrix is performed by finding the eigenvalues and eigenvectors of the matrix. The eigenvectors are then used to form a diagonal matrix, with the eigenvalues as the entries along the diagonal. This transformation is achieved through a process called similarity transformation.
A matrix can be diagonalized if it is a square matrix and has a set of linearly independent eigenvectors. This means that the matrix must have as many distinct eigenvalues as its size, and all of its eigenvectors must be linearly independent.
The eigenvalues and eigenvectors play a crucial role in diagonalization of a matrix. The eigenvalues determine the diagonal entries of the diagonalized matrix, while the eigenvectors form the basis for the transformation. They also provide information about the behavior and properties of the matrix, such as its stability and dynamics.