A singular matrix is a matrix that is not invertible. If a matrix is not invertible, then:
• The determinant of the matrix is 0.
• Any matrix multiplied by that matrix doesn't give the identity matrix.
There are a lot of examples in which a singular matrix is an idempotent matrix. For instance:
M =
[1 1]
[0 0]
Take the product of two M's to get the same M, the given!
M x M = M
So yes, SOME singular matrices are idempotent matrices! How? Let's take a 2 by 2 identity matrix for instance.
I =
[1 0]
[0 1]
I x I = I obviously.
Then, that nonsingular matrix is also idempotent!
Hope this helps!
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A non-singular matrix is basically one that has a multiplicative inverse. More specifically, a matrix "A" is non-singular if there is a matrix "B", such that AB = BA = 1, where "1" is the unity matrix. Non-singular matrixes are those that have a non-zero determinant. Singular and non-singular matrixes are only defined for square matrixes.
There is no reason why it should! So the question is based on an incorrect assumption. A matrix of only zero vectors will be singular!
When its matrix is non-singular.
No. Say your matrix is called A, then a number e is an eigenvalue of A exactly when A-eI is singular, where I is the identity matrix of the same dimensions as A. A-eI is singular exactly when (A-eI)T is singular, but (A-eI)T=AT-(eI)T =AT-eI. Therefore we can conclude that e is an eigenvalue of A exactly when it is an eigenvalue of AT.
It is a singular matrix.