The Chapter 5.15.3 Super Summary
Chapter 5.1.
●
The
eigenvectors
of a matrix
are the
nonzero
vectors
such that
is some
scalar multiple of
.
●
If
is an eigenvector, then its
eigenvalue
is the scalar
where
. The
eigenvalues of a matrix is the set of possible eigenvalues for all its eigenvectors.
●
For any matrix
and any eigenvalue
of
, the
eigenspace
of
corresponding to
is the set of all eigenvectors of
with eigenvalue
. All eigenspaces are vector spaces.
●
If
is triangular (either upper or lower), its eigenvalues are the entries on its main
diagonal. This counts for diagonal matrices as well.
Chapter 5.2.
●
There’s a new way to find the determinant of a square matrix. Use row reductions to put
it in row echelon form, except:
○
Do NOT multiply a row inplace. (You can still add a multiple of a row to another)
○
Keep track of how many times you exchange rows.
Once you finish this, multiply the diagonal terms together (if you end up with a zero in
a diagonal, your matrix is not invertible and has determinant zero). Then, multiply by
, where
is the number of rows exchanges you used. The result will be your
determinant.
○
Alternatively, you could multiply rows inplace, so long as you keep track of the
multiples. For example, if you divide a row by 2 while rowreducing, you should
multiply your final result by 2 to get the determinant. If you divide a row by 3
and multiply another by 4, you’ll have to multiply your result by 3 then divide by
4. Remember, you only need to keep track of this if you multiply a row
inplace
;
adding a multiple of a row to another can be done freely.
○
Also, never multiply a row by zero  that would mean you’d have to divide your
end result by it!
●
The following are equivalent (if one is true, they all are true):
○
The matrix
is not invertible.
○
The number
is an eigenvalue of
○
●
Recall that:
○
○
○
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●
The
characteristic equation
of
is:
is an eigenvalue of
if and only if it’s a solution of the characteristic equation.
●
The number of times a factor appears in the characteristic equation is the
multiplicity
of
the eigenvalue. For example, if this is your characteristic equation:
Then the eigenvalue
has multiplicity 4, the eigenvalue
has multiplicity 2, and the
eigenvalue
has multiplicity 1.
●
The dimension of an eigenspace is, at the
maximum
, the multiplicity of the eigenvalue. It
could be anything between this value and 1.
●
For any
matrices
and
, if there exists an
invertible matrix
such that:
Then
and
are
similar
(some call them
conjugate
).
Similar matrices have the same characteristic equation, and the same eigenvalues (but
not
the same eigenvectors!)
Chapter 5.3.
●
Let
be an
matrix.
is called
diagonalizable
if it’s similar to a diagonal matrix,
i.e. if there exists an
invertible matrix
and an
diagonal matrix
, such
that:
●
The following are all equivalent, for any
matrix
:
○
is diagonalizable.
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 Spring '03
 BUSS
 Linear Algebra, Eigenvectors, Vectors, Scalar, characteristic equation, orthonormal eigenvectors

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