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3. Show that I ® I = I m n mn Describe the matrices I ® A, A ® I. If A is m x m and B is n x n, then A ® B = (A ® I ) (I ® B) = (I ® B) (A ® I ) . n m m n Introductory Matrix Material 24 4. If A and B are upper (or lower) triangular, then so is A 0 B. 5. If A 0 B ^ 0 is diagonal, so are A and B. 6. Let A and B have orders m, n respectively. , n, where A^ and yg are the eigenvalues of A and B. This matrix is often called the Kronecker sum of A and B. 7. Let A and B be of orders m and n. If A and B both are (1) normal, (2) Hermitian, (3) positive definite, (4) positive semidefinite, and (5) unitary, then A 0 B has the corresponding property.

1 w 2 w 1 2 w 4 . 5) T T F = F , F* = (F* ) 1 = F, F = F*. It is of fundamental importance that The Fourier Matrix F is unitary: 1 = F* II or or % Proof. 1, 33 This is a result of the geometric series ny1 wr(j-k)= r=0 n if j = k, ( 11 if j - Wn(j'k) = 1 1 - ^ ' 0 if j t* k. 2 = f *F* = r = 1 0 0 0 0 0 0 II Pm * F * 4 = T2 = I. CO Corollary. 1 \ 1 \ 0 ... 7) F = 7l . (It may be shown that all the qth roots of I are of the form M 1DM where D = diagi]^, p2 , •••* U ) / y? ) Corollary. The eigenvalues of F are ±1, ±i, with appropriate multiplicities.

Dim V = dim L + dim M, L Pi M = {0}. ,y } is a basis for V. x m The fundamental theorem of rank-canonical form for square matrices tells us that if A is a n x n matrix of rank r, then there exist nonsingular matrices P, Q such that PAQ = I © 0 _ . Verify this formulation. 3 KRONECKER PRODUCT Let A and B be m x n and p x q respectively. 1) 11 B' a12B ' •**' aln mlB ' a nB . > a m2 ' ' mn A 0 B = Important properties of the Kronecker product are as follows (indicated operations are assumed to be defined): (1) (aA) ® B = A 0 (aB) = a(A 0 B); a scalar.