So the way this looks in practice is as follows, so let's review, revisit this
example, so we have a matrix here is 2 by 2 and so currently our current goal is
to calculate eigenvalues lambda, okay, and so here is this matrix in this
example, it's a 2 by 2 matrix and there is some blank space here and so we want
to know what are the eigenvalues, right, this is what we want, the eigenvalues of
this 2 by 2. Well, those eigenvalues and the answer is eigenvalues are
-1 and 5, right, but why, okay, that's what we want
to see next, so determining that these are the eigenvalues
is something that comes from the definition of eigenvalues and eigenvectors
which is this vector equation, the matrix multiplied by a nontrivial
vector X equals lambda times identity times X and that is translated into this
is the identity of the 2 by 2 identity so i2 is the 2 by 2 matrix that contains
1 0 in the first column and 0 1 and the second column, okay, so this would be
considered I sub 2, that's the two by two identity, so even though we're
interested on lambda which for which this vector equation is true, by
exploring the null space of a minus lambda I, specifically by making use of
the fact that if the matrix is singular the determinant is 0, so we say we'll
consider the determinant of A minus lambda equal to 0, and this will be considered
the determinant of A minus lambda times I, okay, and so this becomes A minus lambda I is
this matrix here, this matrix is 2 by 2 and that is a minus lambda i,
which is basically take the original matrix and subtract lambda from
the diagonal entries. You're still operating with a 2 a 2 matrix, we're
subtracting lambda from from the diagonal entries but now this whole
thing is a scalar equation, okay, because the determinant is an operator, the, while
that, just remember the determinant is an operator, the input of this operator is
matrix, the output of that operator is a scalar, so we compute the determinant of
a matrix and that is on its own a scalar and the determinant of this matrix
involves multiplying the diagonal entries and subtracting, so here we
multiply -1 minus lambda times 5 minus lambda and subtract 0
times 2, that is 0, right, so we come up with this expression minus lambda, which
is -1 minus lambda, 5 minus lambda and so what happens this step here is
that remember, all these left-hand sides, they, we carry out, right, we start with 0
equals determinant, we're working our way through the right side with that first
expression, right, and now we arrive at expression 0 equals this product here,
right, 0 equals to that product, so what happens if we multiply both sides by
-1, we multiply both sides by -1, we obtain this and from that
expression, so maybe some little bit more of arithmetic or algebra we need to do
here so from this expression we have 0 equals lambda 1 times quantity lambda
minus 5, right, and from this expression we say, you know, this is true if and only
if lambda plus 1 equals 0 or lambda minus 5 equals 0, so from lambda plus 1
equals 0 we say that is true if and only if lambda equals minus 1 and then from
this expression we say lambda minus five equals zero if and only if lambda equals
five, so we find two potential values here, that's all these equations, and
these values are precisely the value of -1 and five and these are the
reasons, okay? That's how we arrive at specific values
of lambda. We started by assuming that zero is equal to the determinant of A
minus lambda I, this is a direct connection to the definition of
singularity, if the matrix A minus lambda I is singular then the determinant of
that matrix is zero.