Section 7.2
... By use of similarity transformations, we change a matrix A into a similar matrix which is “simpler” in its form. These are often called canonical forms; and there is a large literature on such forms. The ones most used in numerical analysis are: The Schur normal form The principal axes form The Jord ...
... By use of similarity transformations, we change a matrix A into a similar matrix which is “simpler” in its form. These are often called canonical forms; and there is a large literature on such forms. The ones most used in numerical analysis are: The Schur normal form The principal axes form The Jord ...
Matrix multiplication and composition of linear
... 1. There exists an n × p matrix B such that T = TB , i.e., such that T (X) = TB (X) for all X ∈ Rp . 2. T satisfies the principle(s) of superposition: (a) T (X + X 0 ) = T (X) + T (X 0 ) for all X and X 0 in Rp , and (b) T (cX) = cT (X) for all X ∈ Rp and c ∈ R. Proof: The proof that (1) implies (2) ...
... 1. There exists an n × p matrix B such that T = TB , i.e., such that T (X) = TB (X) for all X ∈ Rp . 2. T satisfies the principle(s) of superposition: (a) T (X + X 0 ) = T (X) + T (X 0 ) for all X and X 0 in Rp , and (b) T (cX) = cT (X) for all X ∈ Rp and c ∈ R. Proof: The proof that (1) implies (2) ...
5.6 UNITARY AND ORTHOGONAL MATRICES
... Conversely, if (5.6.1) holds, then U must be unitary. To see this, set x = ei in (5.6.1) to observe u∗i ui = 1 for each i, and then set x = ej + ek for j = k to obtain 0 = u∗j uk + u∗k uj = 2 Re (u∗j uk ) . By setting x = ej + iek in (5.6.1) it also follows that 0 = 2 Im (u∗j uk ) , so u∗j uk = 0 f ...
... Conversely, if (5.6.1) holds, then U must be unitary. To see this, set x = ei in (5.6.1) to observe u∗i ui = 1 for each i, and then set x = ej + ek for j = k to obtain 0 = u∗j uk + u∗k uj = 2 Re (u∗j uk ) . By setting x = ej + iek in (5.6.1) it also follows that 0 = 2 Im (u∗j uk ) , so u∗j uk = 0 f ...
