Find the orthogonal canonical reduction of the quadratic form x2 - 2y2 + z2 + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.
The given quadratic form is Q = x² − 2y² + z² + 2xy + 6yz. It can be written in matrix form as Q = [x y z] A [x y z]ᵀ, where A = _______ ____ _________ ____ ___ _____ _________ ____ _________.
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Check that for any
.
Find the values of for which the matrix
is Hermitian.
) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) The function defined by
is 1-1.
ii) The operation $ defined byx y = \log(xy)is a binary operation onS, whereSis the set\{x \in \mathbf{R} \mid x > 0\}$.
iii) The set is a subspace of
.
iv) There is no matrix of rank 6.
v) If V and V' are vector spaces and is a linear transformation, then whenever
are linearly independent,
are also linearly independent.
vi) If V is a vector space and is a linear operator with
, then T is not diagonalisable.
vii) The degree of the minimal polynomial of a matrix is at most 2.
viii) For any matrix A,
.
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form x2 + y2 + z2 to the quadratic form xz + yz.
Show that, if A is any matrix with real entries, then there is a
symmetric matrix S and a
skew symmetric matrix S' such that
.
For the following matrices, check whether there exists an invertible matrix P such that P-1AP is diagonal. When such a P exists, find P.
i)
Find the orthogonal canonical reduction of the quadratic form x2 - 2y2 + z2 + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.
Find the direction cosines of the perpendicular from the origin to the plane