2.5 Multiplication of Matrices Outline Multiplication of
... previous section that addition, subtraction, and scalar multiplication are done entrywise. However, multiplication of matrices is not done entrywise. It turns out that when we are dealing with data matrices, entrywise multiplication often does not give us the information we are looking for. Given tw ...
... previous section that addition, subtraction, and scalar multiplication are done entrywise. However, multiplication of matrices is not done entrywise. It turns out that when we are dealing with data matrices, entrywise multiplication often does not give us the information we are looking for. Given tw ...
ch1.3 relationship between IO and state space desicriptions
... 2) To prove that any solution can be expressed as the linear combination of the n solutions. That is, all the solutions of (1-50) form an ndimensional vector space. Let (t) be an arbitrary solution of (1-50) with ...
... 2) To prove that any solution can be expressed as the linear combination of the n solutions. That is, all the solutions of (1-50) form an ndimensional vector space. Let (t) be an arbitrary solution of (1-50) with ...
Review of Linear Algebra
... We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of Rn , which here we will denote with bold face letters such as v, and scalars, in other words elements of R. We could also use Cn , or Qn as neede ...
... We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of Rn , which here we will denote with bold face letters such as v, and scalars, in other words elements of R. We could also use Cn , or Qn as neede ...
matrices and systems of equations
... • Write matrices and identify their orders. • Perform elementary row operations on matrices. • Use matrices and Gaussian elimination to solve systems of linear equations. • Use matrices and Gauss-Jordan elimination to solve systems of linear equations. ...
... • Write matrices and identify their orders. • Perform elementary row operations on matrices. • Use matrices and Gaussian elimination to solve systems of linear equations. • Use matrices and Gauss-Jordan elimination to solve systems of linear equations. ...
Multiplication of Matrices
... and x = C●,k is a column vector. On the other hand by the first definition of matrix multiplication ((AB)C)ik = ((AB)i,●)(C●,k). By the third definition of matrix multiplication (AB)i,● = (Ai,●)B. So ((AB)C)ik = ((Ai,●)B)(C●,k) = (pB)x. However, in the previous section we proved that if p is a row v ...
... and x = C●,k is a column vector. On the other hand by the first definition of matrix multiplication ((AB)C)ik = ((AB)i,●)(C●,k). By the third definition of matrix multiplication (AB)i,● = (Ai,●)B. So ((AB)C)ik = ((Ai,●)B)(C●,k) = (pB)x. However, in the previous section we proved that if p is a row v ...
Solution Set
... Therefore Wv is a subspace of R2 . Now assume that there exists a subspace V not of the form Wv . It must contain at least two vectors v and w, where w 6= cv. Then verifying property (M1), we must have that cv + dw ∈ V for any real constants c and d. This means that vectors in V are generated by two ...
... Therefore Wv is a subspace of R2 . Now assume that there exists a subspace V not of the form Wv . It must contain at least two vectors v and w, where w 6= cv. Then verifying property (M1), we must have that cv + dw ∈ V for any real constants c and d. This means that vectors in V are generated by two ...
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.