4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular
... Matrices may be added, subtracted, and multiplied, provided their dimensions satisfy certain restrictions. To add or subtract two matrices, the matrices must have the same dimensions. Notice there are two types of multiplication. Scalar multiplication refers to the product of a matrix times a scalar ...
... Matrices may be added, subtracted, and multiplied, provided their dimensions satisfy certain restrictions. To add or subtract two matrices, the matrices must have the same dimensions. Notice there are two types of multiplication. Scalar multiplication refers to the product of a matrix times a scalar ...
3 The positive semidefinite cone
... > 0 such that kA − Xk ≤ ⇒ X ∈ Sn+ . Let X = A − I where I is the n × n identity matrix, and note that kA − Xk = kIk ≤ . It thus follows that X = A − I ∈ Sn+ . Since the eigenvalues of A − I are the (λi − ) (where (λi ) are the eigenvalues of A) it follows that λi ≥ > 0 and thus A is posi ...
... > 0 such that kA − Xk ≤ ⇒ X ∈ Sn+ . Let X = A − I where I is the n × n identity matrix, and note that kA − Xk = kIk ≤ . It thus follows that X = A − I ∈ Sn+ . Since the eigenvalues of A − I are the (λi − ) (where (λi ) are the eigenvalues of A) it follows that λi ≥ > 0 and thus A is posi ...
Complex inner products
... There is a complex version of orthogonal matrices. A complex square matrix U is called unitary if U ∗ = U −1 . Equivalently, the columns of U form an orthonormal set (using the standard Hermitian inner product on Cn ). Any orthogonal matrix is unitary. Likewise, there is a complex version of symmetr ...
... There is a complex version of orthogonal matrices. A complex square matrix U is called unitary if U ∗ = U −1 . Equivalently, the columns of U form an orthonormal set (using the standard Hermitian inner product on Cn ). Any orthogonal matrix is unitary. Likewise, there is a complex version of symmetr ...
10.2. (continued) As we did in Example 5, we may compose any two
... a rotation, a translation, a reflection, or a glide reflection. Let us first record two consequences of this theorem. Corollary 1. The composition of any two rotations, or of any two reflections, must be a rotation, a translation or the identity. Corollary 2. Any isometry f : E → E is a bijective ma ...
... a rotation, a translation, a reflection, or a glide reflection. Let us first record two consequences of this theorem. Corollary 1. The composition of any two rotations, or of any two reflections, must be a rotation, a translation or the identity. Corollary 2. Any isometry f : E → E is a bijective ma ...
Lecture 28: Eigenvalues - Harvard Mathematics Department
... as the other patterns only give us terms which are of order λn−2 or smaller. How many eigenvalues do we have? For real eigenvalues, it depends. A rotation in the plane with an angle different from 0 or π has no real eigenvector. The eigenvalues are complex in that case: ...
... as the other patterns only give us terms which are of order λn−2 or smaller. How many eigenvalues do we have? For real eigenvalues, it depends. A rotation in the plane with an angle different from 0 or π has no real eigenvector. The eigenvalues are complex in that case: ...
= 0. = 0. ∈ R2, B = { B?
... (b) It is easy to see (and it may be formally proved by induction) that T k α j = α j+k if j + k ≤ n, and T k α j = 0 if j + k > n. Hence, T n α j = 0 for 1 ≤ j ≤ n, so since the α j form a basis, T n = 0. Also, T n−1 α1 = αn , so T n−1 6= 0. (c) Since Sn−1 6= 0, there exists v ∈ V such that Sn−1 v ...
... (b) It is easy to see (and it may be formally proved by induction) that T k α j = α j+k if j + k ≤ n, and T k α j = 0 if j + k > n. Hence, T n α j = 0 for 1 ≤ j ≤ n, so since the α j form a basis, T n = 0. Also, T n−1 α1 = αn , so T n−1 6= 0. (c) Since Sn−1 6= 0, there exists v ∈ V such that Sn−1 v ...