6. True or false (10 points, no need (1) Any elementary matrix has a QR decompos
ID: 3149089 • Letter: 6
Question
6. True or false (10 points, no need (1) Any elementary matrix has a QR decomposit (2) Any elemen 2yary matrix has a L.U decompoition. to explain your reasoning) (3) If -P then P is (El Th must benorthogonaquare matrix are orthogonal complements of cach other, the (5) The charactcristic polynomial of a 4 x4 symmetric matrix must bae an orthogonal projection matrix. tho null space and column space of a are orthog matrix must be an orthogonal projection matrix. I the characteristic polynomial of A is AeeP-I) and A is diagonaliza ble, then the rank of A is 3 )The dimension of the orthogonal complencut of the column space of a square matrix is the dimeusion of its null space. (8) An orthogonal projection matrix must be orthogonal. (9) The reduced row echelon form of an orthogonal matrix is the identity matrx A, then if A is symmetric so is B.Explanation / Answer
6
(1) True. Every elementary matrix is essentially a square matrix. Every square matrix has a QR decomposition.
(2) False. Elementary matrices are identity matrix acted upon by any one elementary operation i.e row switching, multiplication or addition. In row switching (permutation matrices) the elementary cannot be LU decomposed. It can be decomposed as PLU(Permutation Lower Upper).
(3) When P2 = P, it is just a projection matrix, need mot be orthogonal. In fact it is the definition of a projection matrix.
(5) All symmetric matrices have real eigen values. The roots of characteristic polynomials are eigen values. Therefore all roots are real. And by fundamental theorem of algebra the charateristic polynomial will have four real roots
(6) True. The dimension of the matrix is 5 . Since the matrix has a double root at zero, there exists two vectors v1, v2 such that Avi = 0. Therefore the null space has dimension 2. By rank nullity theorem, the rank of the matrix is 3.
(8) False. Orthogonal projection matrices can be singular. The projection from (x,y,z) pane to (x,y,0) plane is orthogonal. But the matrix is not.
(9) True. The row echelon form can have non zero determinant only if it is identity. The determinant of an orthogonal matrix is 1 or -1. Therefore its row echelon form should also have non zero determinant.
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