This is a group because the product of any two scalar multiples of the identity matrix is another scalar multiple of the identity matrix, and the product of any powers of 2 is another power of 2.
22.
This is a group because the product of any two scalar multiples of the identity matrix is another scalar multiple of the identity matrix, and the product of any powers of 2 is another power of 2.
23.
A positive scalar multiple of a solution is also a solution, so you could convert this in various ways to a question about affine constraints in \ mathbb { R } ^ { m-1 }.
24.
The "'classification theorem "'for C 0 contractions states that two multiplicity free C 0 contractions are quasi-similar if and only if they have the same minimal function ( up to a scalar multiple ).
25.
This is somewhat elementary, and perhaps you know this already . but in the special case that all your blocks are scalar multiples of each other, then it is a Kronecker product and you can get the eigenvalues of the big matrix.
26.
To prove this result one notes first that a representation is irreducible if and only if the commutant of ? ( " A " ), denoted by ? ( " A " )', consists of scalar multiples of the identity.
27.
If q _ 0 + iq _ 1 + jq _ 2 + kq _ 3 is not a unit quaternion then the homogeneous form is still a scalar multiple of a rotation matrix, while the inhomogeneous form is in general no longer an orthogonal matrix.
28.
This shows that ? is irreducible if and only if any such ? " g " is unitarily equivalent to ?, i . e . " g " is a scalar multiple of " f ", which proves the theorem.
29.
A function that is Fr�chet differentiable at a point is necessarily continuous there and sums and scalar multiples of Fr�chet differentiable functions are differentiable so that the space of functions that are Fr�chet differentiable at a point form a subspace of the functions that are continuous at that point.
30.
Since the space of modular forms of weight 2 " k " has dimension 1 for 2 " k " = 4, 6, 8, 10, 14, different products of Eisenstein series having those weights have to be equal up to a scalar multiple.
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