Notes - Linear Algebra II HT23, Determinants

[[Course - Linear Algebra II HT23]]^U
- [[Notes - Linear Algebra II HT23, Permutations]]^U

Determinants

When building up determinants as a determinantal mapping $D : M_{n} (R) \to R$ , what are the three conditions a determinantal map must satsify?

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_i, a a_i + 1, \dots] = 0 when a a_i = a a_i + 1

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

What’s the geometric interpretation behind a determinantal map being multilinear in the columns, i.e. $D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]$ ?

It represents the fact that stretching a face with scale the area/volume accordingly.

What’s the geometric interpretation behind a determinantal map being alternating, i.e. $D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}$ ?

A shape with thickness has no volume.

If $λ = det [\dots, a a, \dots, b b, \dots]$ , then what is $det [\dots, b b, \dots, a a, \dots]$ ?

- λ

What is $det [\dots, a a, \dots, a a, \dots]$ ?

0

When proving that there exists a map $D_{n}$ that satisfies the properties of being a determinantal map on $n \times n$ matrices

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

In the inductive step, what do you define $D_{n}$ as, in terms of $D_{n - 1}$ ?

D_{n} (A) = a_{11} D_{n - 1} (A_{11}) - a_{12} D_{n - 2} (A_{12}) + \dots + (- 1)^{n - 1} a_{1 n} D_{n - 1} (A_{1 n})

Can you give the Laplace expansion formula for $det A$ along row $i$ ?

det A = \sum_{j = 1}^{n} a_{i j} (- 1)^{i + j} det A_{i j}

Can you give the Laplace expansion formula for $det A$ along column $j$ ?

det A = \sum_{i = 1}^{n} a_{i j} (- 1)^{i + j} det A_{i j}

Can you give the permutation formula for $det A$ ?

det A = \sum_{σ perm. of 1, \dots, n} sgn (σ) a_{σ (1) 1} a_{σ (2) 2} \dots a_{σ (n) n}

What is $det A^{⊺}$ ?

det A

What is $det B A$ ?

det A det B

Why is $det A$ multilinear in rows as well as columns, despite this not being in the definition?

Because $det A = det A^{⊺}$ .

Using the Rule of Sarrus (cool name) mneumonic, what is

| \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} |

\begin{aligned} a e i + d h c + g b f \\ - & c e g - f h a - i b d \end{aligned}

If $A$ is a upper or lower triangular matrix, what is the determinant?

\prod_{i = 1}^{n} a_{i i}

To prove the more general $det A B = det A det B$ , what do you show first?

det E A = det E det A

where $E$ is an elementary matrix.

How does the determinant change when you swap two rows of a matrix?

It switches sign.

How does the determinant change when you add a scalar multiple of one row to another in a matrix?

It does not change.

How does the determinant change when you multiply one row of a matrix by a non-zero scalar $λ$ ?

The determinant is multiplied by $λ$ .

Proofs

Important ones

Prove that there exists a map $D_{n}$ that satisfies the properties of being a determinantal map on $n \times n$ matrices.

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

Todo?

Prove if there exists a unique map $D_{n}$ that satisfies the properties of being a determinantal map on $n \times n$ matrices:

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

Hint: This involves showing the following formula is true for the determinant:

det A = \sum_{σ perm. of 1, \dots, n} sgn (σ) a_{σ (1) 1} a_{σ (2) 2} \dots a_{σ (n) n}

Todo?

Not-so-important ones

Prove that if a map $D_{n}$ satisfies the properties of being a determinantal map on $n \times n$ matrices

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

Then the alternating condition can be strengthened to

det [\dots, a a, \dots, a a, \dots] = 0

Todo?

Prove that if a map $D_{n}$ satisfies the properties of being a determinantal map on $n \times n$ matrices

Multilinear in the columns

D [\dots, λ b b_{i} + μ c c_{i}, \dots] = λ D [\dots, b b_{i}, \dots] + μ D [\dots, a a_{i}, \dots]

Alternating

D [\dots, a a_{i}, a a_{i + 1}, \dots] = 0 when a a_{i} = a a_{i + 1}

$D (I_{n}) = 1$ for the $n \times n$ identity matrix

Then it is also true that

det [\dots, a a, \dots, b b, \dots] = - det [\dots, b b, \dots, a a, \dots]

Todo?

Prove that $det A = det A^{⊺}$ .

Todo.

Prove

det A = 0 ⟺ A singular

Todo.

Prove

det E A = det E det A

where $E$ is an elementary matrix.

Todo

Prove

det A B = det A det B

by appealing to a lemma.

Todo.

Prove that if $V$ is a vector space and $T : V \to V$ is linear then $det T$ defined by $det T = det M^{B}$ where $M^{B}$ is the transformation matrix of $T$ with respect to a basis $B$ does not depend on the choice of basis.

Todo.

Determinants

Proofs

Important ones

Not-so-important ones

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