Diagonalization
| Description: This quiz is designed to assess your understanding of the concept of diagonalization in linear algebra. | |
| Number of Questions: 15 | |
| Created by: Aliensbrain Bot | |
| Tags: linear algebra diagonalization eigenvalues eigenvectors |
What is the process of finding a matrix that is similar to a given matrix called?
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Diagonalization
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Triangularization
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Orthogonalization
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Jordanization
Diagonalization is the process of finding a matrix that is similar to a given matrix, meaning that they have the same eigenvalues and eigenvectors.
What is an eigenvalue of a matrix?
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A scalar that, when multiplied by the matrix, produces the matrix itself
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A scalar that, when multiplied by the matrix, produces the zero matrix
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A scalar that, when multiplied by the matrix, produces a diagonal matrix
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A scalar that, when multiplied by the matrix, produces an orthogonal matrix
An eigenvalue of a matrix is a scalar that, when multiplied by the matrix, produces a diagonal matrix.
What is an eigenvector of a matrix?
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A vector that, when multiplied by the matrix, produces the eigenvalue of the matrix
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A vector that, when multiplied by the matrix, produces the zero vector
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A vector that, when multiplied by the matrix, produces a diagonal matrix
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A vector that, when multiplied by the matrix, produces an orthogonal matrix
An eigenvector of a matrix is a vector that, when multiplied by the matrix, produces the eigenvalue of the matrix.
What is the diagonalization theorem?
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A theorem that states that every square matrix can be diagonalized
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A theorem that states that every square matrix has at least one eigenvalue
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A theorem that states that every square matrix has at least one eigenvector
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A theorem that states that every square matrix is similar to a diagonal matrix
The diagonalization theorem states that every square matrix is similar to a diagonal matrix.
What is the Jordan canonical form of a matrix?
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A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal
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A matrix that is similar to a given matrix and has all of its eigenvectors as its columns
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A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its columns
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A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its rows
The Jordan canonical form of a matrix is a matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal.
What is the characteristic polynomial of a matrix?
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A polynomial whose roots are the eigenvalues of the matrix
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A polynomial whose roots are the eigenvectors of the matrix
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A polynomial whose roots are the diagonal elements of the matrix
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A polynomial whose roots are the off-diagonal elements of the matrix
The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix.
What is the minimal polynomial of a matrix?
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A polynomial that is the lowest-degree polynomial that annihilates the matrix
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A polynomial that is the lowest-degree polynomial that has the matrix as a root
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A polynomial that is the lowest-degree polynomial that has the eigenvalues of the matrix as its roots
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A polynomial that is the lowest-degree polynomial that has the eigenvectors of the matrix as its roots
The minimal polynomial of a matrix is a polynomial that is the lowest-degree polynomial that annihilates the matrix.
What is the Cayley-Hamilton theorem?
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A theorem that states that every square matrix satisfies its own characteristic polynomial
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A theorem that states that every square matrix satisfies its own minimal polynomial
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A theorem that states that every square matrix is similar to a diagonal matrix
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A theorem that states that every square matrix has at least one eigenvalue
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial.
What is the Schur decomposition of a matrix?
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A decomposition of a matrix into a product of two unitary matrices
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A decomposition of a matrix into a product of two orthogonal matrices
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A decomposition of a matrix into a product of two diagonal matrices
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A decomposition of a matrix into a product of two triangular matrices
The Schur decomposition of a matrix is a decomposition of a matrix into a product of two unitary matrices.
What is the singular value decomposition of a matrix?
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A decomposition of a matrix into a product of three matrices
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A decomposition of a matrix into a product of four matrices
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A decomposition of a matrix into a product of five matrices
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A decomposition of a matrix into a product of six matrices
The singular value decomposition of a matrix is a decomposition of a matrix into a product of three matrices.
What is the QR decomposition of a matrix?
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A decomposition of a matrix into a product of two matrices
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A decomposition of a matrix into a product of three matrices
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A decomposition of a matrix into a product of four matrices
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A decomposition of a matrix into a product of five matrices
The QR decomposition of a matrix is a decomposition of a matrix into a product of two matrices.
What is the LU decomposition of a matrix?
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A decomposition of a matrix into a product of two matrices
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A decomposition of a matrix into a product of three matrices
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A decomposition of a matrix into a product of four matrices
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A decomposition of a matrix into a product of five matrices
The LU decomposition of a matrix is a decomposition of a matrix into a product of two matrices.
What is the Cholesky decomposition of a matrix?
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A decomposition of a positive-definite matrix into a product of two triangular matrices
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A decomposition of a positive-definite matrix into a product of three triangular matrices
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A decomposition of a positive-definite matrix into a product of four triangular matrices
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A decomposition of a positive-definite matrix into a product of five triangular matrices
The Cholesky decomposition of a matrix is a decomposition of a positive-definite matrix into a product of two triangular matrices.
What is the eigenvalue-eigenvector method for solving a system of linear differential equations?
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A method for solving a system of linear differential equations by finding the eigenvalues and eigenvectors of the coefficient matrix
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A method for solving a system of linear differential equations by finding the characteristic polynomial of the coefficient matrix
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A method for solving a system of linear differential equations by finding the minimal polynomial of the coefficient matrix
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A method for solving a system of linear differential equations by finding the Cayley-Hamilton theorem of the coefficient matrix
The eigenvalue-eigenvector method for solving a system of linear differential equations is a method for solving a system of linear differential equations by finding the eigenvalues and eigenvectors of the coefficient matrix.
What is the power method for finding the largest eigenvalue and corresponding eigenvector of a matrix?
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A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a random vector
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A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is orthogonal to the previous vector
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A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is parallel to the previous vector
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A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is equal to the previous vector
The power method for finding the largest eigenvalue and corresponding eigenvector of a matrix is a method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a random vector.