0
Yes it is. In fact, every singular operator (read singular matrix) has 0 as an eigenvalue (the converse is also true). To see this, just note that, by definition, for any singular operator A, there exists a nonzero vector x such that Ax = 0. Since 0 = 0x we have Ax = 0x, i.e. 0 is an eigenvalue of A.
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LaoWell, hello there! It's okay to wonder about eigenvalues. Zero can indeed be an eigenvalue. It simply means that when a matrix is applied to a vector, the resulting vector is in the same direction as the original vector, but possibly scaled by zero. Just a happy little mathematical concept to explore!
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Prove that a matrix a is singular if and only if it has a zero eigenvalue?
Recall that if a matrix is singular, it's determinant is zero. Let our nxn matrix be called A and let k stand for the eigenvalue. To find eigenvalues we solve the equation det(A-kI)=0for k, where I is the nxn identity matrix. (<==) Assume that k=0 is an eigenvalue. Notice that if we plug zero into this equation for k, we just get det(A)=0. This means the matrix is singluar. (==>) Assume that det(A)=0. Then as stated above we need to find solutions of the equation det(A-kI)=0. Notice that k=0 is a solution since det(A-(0)I) = det(A) which we already know is zero. Thus zero is an eigenvalue.
What are the factors of 0?
0 has no factors.
What is the difference between the zero property of multiplication and the identity property of addition?
Usually, the identity of addition property is defined to be an axiom (which only specifies the existence of zero, not uniqueness), and the zero property of multiplication is a consequence of existence of zero, existence of an additive inverse, distributivity of multiplication over addition and associativity of addition. Proof of 0 * a = 0: 0 * a = (0 + 0) * a [additive identity] 0 * a = 0 * a + 0 * a [distributivity of multiplication over addition] 0 * a + (-(0 * a)) = (0 * a + 0 * a) + (-(0 * a)) [existence of additive inverse] 0 = (0 * a + 0 * a) + (-(0 * a)) [property of additive inverses] 0 = 0 * a + (0 * a + (-(0 * a))) [associativity of addition] 0 = 0 * a + 0 [property of additive inverses] 0 = 0 * a [additive identity] A similar proof works for a * 0 = 0 (with the other distributive law if commutativity of multiplication is not assumed).
What is the answer for 0 divided by 7?
0
What is an eigenvector?
Oh, dude, an eigenvector is like a fancy term in math for a vector that doesn't change direction when a linear transformation is applied to it. It's basically a vector that just chills out and stays the same way, no matter what you do to it. So, yeah, eigenvectors are like the cool, laid-back dudes of the math world.
