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Title: Linear Algebra - Determinants and Eigenvalues
Description: These notes cover all of Determinants and Eigenvalues
Description: These notes cover all of Determinants and Eigenvalues
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Determinants, Finished
We ended with det(AB) = det(A) det(B)
...
The effect on det(A) of a Row Operation on A
• Row Interchange on A converts det(A) to − det(A)
...
• Row Addition does nothing to the determinant
...
You only need to
construct a matrix B such that BA would be equal to A with the row operation applied
...
0 0 1
Note that det(B) = −1
...
Example:
5 −1
8 −4
2
1 −2
1
3 −1
4 −1
−1
1
0 −1
−1
1
0 −1
2
1 −2
1
=−
3 −1
4 −1
5 −1
8 −4
−1
0
=−
0
0
1
0 −1
3 −2
1
2
4 −4
4
8 −9
−1
0
=2
0
0
1
0 −1
1
2 −2
3 −2
1
4
8 −9
−1
0
=2
0
0
1
0 −1
1
2 −2
...
Property: The determinant of a triangular matrix (either upper or lower triangular) is just
the product of the diagonal
...
det(AT ) = det(A)
...
Example:
[
A=
1 0
0 0
]
[
,
B=
0 0
0 1
]
has det(A + B) = 0
det(A) + det(B) = 0
...
As a result: if A, B are Row Equivalent then
det(A) = 0 ⇐⇒ det(B) = 0 ,
det(A) ̸= 0 ⇐⇒ det(B) ̸= 0
...
So:
det(A) det(A−1 ) = det(AA−1 ) = det(I) = 1
Property: det(A−1 ) =
1
...
[
]
Furthermore, if A is not invertible then (using the A | I algorithm) does not reduce
to I on the left
...
As a result:
A not invertible
=⇒
det(A) = 0
A not invertible
⇐⇒
det(A) = 0
...
We can also say that det(A) = 0 if and only if A has a missing pivot (as in, not n of
them) etc
...
The matrix A simply RE-SCALES (re-sizes, etc) the vector v
...
0
0
0
0
1 0 −2
−2
3 has λ1 = −1 for v1 = 5 :
Example: 2 1
−2 0
1
−2
1 0 −2
−2
2
−2
2 1
3 5 = −5 = (−1) 5
...
So, ANY REAL MULTIPLE of v will also be an eigenvector
...
3
How To Find Them:
If λ is an eigenvalue then we have v ̸= 0 so that
Av = λv
Av − λv = 0
...
Null(A − λI)
non-trivial
=⇒
(A − λI)
not-invertible
...
Note that, as usual, the determinant outputs a number
...
Definition: The polynomial det(A − λI) is the CHARACTERISTIC POLYNOMIAL of A
...
• The FACTORS of the char
...
are the eigenvalues of A
...
Calculate the Characteristic Polynomial of A (det(A − λI))
2
...
poly
...
Find bases for the null spaces of A − λI, with λ the eigenvalues
...
1
0
Example: A =
0
0
triangular matrix (upper or lower) then its eigenvalues are simply its
0 −2 3
9 −2 0
has eigenvalues 1,9,0 and 12
...
1 0 −2
3 (recall that we already have −1)
...
= (1 − λ)
This leads to (1 − λ)(λ − 3)(λ + 1) with factors 1, −1, 3
...
To find the eigenvectors, simply check the null spaces of A − λI, with λ an eigenvalue
...
I
...
0 0 −2 0
3 0
For λ1 = 1 :
solve 2 0
−2 0
0 0
0
1
...
2
−2
For λ3 = −1 we had 5
...
0 0 0 0
1 0
1 0
0 1 −1 0
2
0 0
0 0
Example Questions:
Section 2
...
b), 10
...
bd)
The eigenvalue section in the text mostly covers the next subject, diagonalization
...
bdf), but only find the eigenvalues/vectors
Title: Linear Algebra - Determinants and Eigenvalues
Description: These notes cover all of Determinants and Eigenvalues
Description: These notes cover all of Determinants and Eigenvalues