Multiplying and Factoring Matrices
... The cases of failure are important too (or adjustment more than failure). A = LU now requires an “echelon form E” and diagonalization needs a “Jordan form J”. Matrix multiplication is still columns times rows. A = CE = (m by r)(r by n). ...
... The cases of failure are important too (or adjustment more than failure). A = LU now requires an “echelon form E” and diagonalization needs a “Jordan form J”. Matrix multiplication is still columns times rows. A = CE = (m by r)(r by n). ...
Multilinear spectral theory
... Adopt matrix terminology and call it characteristic polynomial of A, which has an expression pA(λ) = ...
... Adopt matrix terminology and call it characteristic polynomial of A, which has an expression pA(λ) = ...
MATH10212 • Linear Algebra • Examples 2 Linear dependence and
... of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a dependence relationship (coefficients) among the vectors. ...
... of these, the answer can be determined by inspection (i.e., without calculation), state why. For any sets that are linearly dependent, find a dependence relationship (coefficients) among the vectors. ...
Square root sf the Boolean matrix J
... square root of J - I and established the following theorem. The Boolean matrix J - 1 has a square root if and only if its dimension is at least 7 or is I. The author has found out the exact square root, when the dimension is a prime p of the form 4k + 3 K > 1 or when the dimension is 20, I), p = 4k ...
... square root of J - I and established the following theorem. The Boolean matrix J - 1 has a square root if and only if its dimension is at least 7 or is I. The author has found out the exact square root, when the dimension is a prime p of the form 4k + 3 K > 1 or when the dimension is 20, I), p = 4k ...
1 Vector Spaces
... (λ is necessarily an eigenvalue of L), so that the eigenvectors of L together with these generalized eigenvectors form a basis for X and the matrix of L with respect to this basis is in Jordan canonical form. Notice that if L possesses generalized eigenvectors of level m ≥ 2, associated to some eige ...
... (λ is necessarily an eigenvalue of L), so that the eigenvectors of L together with these generalized eigenvectors form a basis for X and the matrix of L with respect to this basis is in Jordan canonical form. Notice that if L possesses generalized eigenvectors of level m ≥ 2, associated to some eige ...
Notes
... about A is a subroutine that can apply A to a vector. What should we do? If the only operation at hand is matrix mutiplication, and the only vector staring us in the face is b, a natural approach would be to take b and start multiplying by A to get b, Ab, A2 b, etc. Taking linear combinations of the ...
... about A is a subroutine that can apply A to a vector. What should we do? If the only operation at hand is matrix mutiplication, and the only vector staring us in the face is b, a natural approach would be to take b and start multiplying by A to get b, Ab, A2 b, etc. Taking linear combinations of the ...
best upper bounds based on the arithmetic
... Theorem : The best bound of the form (4) is obtained when Φ(x) = τ (x) ≡ (2x − 1) + ...
... Theorem : The best bound of the form (4) is obtained when Φ(x) = τ (x) ≡ (2x − 1) + ...
3DROTATE Consider the picture as if it were on a horizontal
... rectangle locate on the Z=0 plane with its upper left corner at the origin. This means that points on the picture have values of (X,Y,0) Consider the perspective transformation as if it were the process of capturing a picture by a frame camera located a distance Zc=f above the picture, where f is th ...
... rectangle locate on the Z=0 plane with its upper left corner at the origin. This means that points on the picture have values of (X,Y,0) Consider the perspective transformation as if it were the process of capturing a picture by a frame camera located a distance Zc=f above the picture, where f is th ...
EC220 - Web del Profesor
... Two vectors can be multiplied if and only if the number of columns of a ROW vector equals the number of rows of a COLUMN vector (both vectors are compatible). By the same token, two matrices A and B can be multiplied if and only if the number of columns of A equals the number of rows of B (Both mat ...
... Two vectors can be multiplied if and only if the number of columns of a ROW vector equals the number of rows of a COLUMN vector (both vectors are compatible). By the same token, two matrices A and B can be multiplied if and only if the number of columns of A equals the number of rows of B (Both mat ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.