Square Matrices

These are matrices over some field $\mathbb{F}$ of size $n\times n$. There are exactly 2 types of square matrices: endomorphisms and bilinear forms.

Theorem (Operations on Square Matrices)

You can view eigenvalues, trace, and the determinant.

Definition (Endomorphisms)

Linear Maps from $V$ to itself are called endomorphisms.

$$\text{End}(V)=\left\{f:V\stackrel{\text{ linear}}{\to }V\right\}$$

Via index notation, we use $A_i^j$ to denote the matrix representation of an endomorphism.

Definition (Bilinear Forms)

A bilinear form on $V$ are linear maps $B:V\to V^{\ast }$. A bilinear form works like

$$\underset{\in V^{\ast }}{\underbrace{B(u)}}(v)$$

like a bilinear function on $V$. Via index notation, we use $B_{ij}$ to denote the matrix representation of a bilinear form.

Remark (Endomorphisms vs Bilinear Forms)

Operations on matrices such as taking trace, determinant, and eigenvalues; or properties of being identity, etc. makes sense only for endomorphisms.

Eigenvalues, trace, and determinants of $A_1{A}_2\dots A_k$ is invariant under cyclic permutations of the $A_i$‘s.

$$\text{tr}(A_1{A}_2\dots A_k)=\text{tr}(A_k{A}_1{A}_2\dots A_{k-1})=\cdots$$

Remark

Properties on matrices such as being symmetric (self-adjoint) makes sense only for bilinear forms.

$$B\in \left(V\stackrel{\text{ linear}}{\to }{V}^{\ast }\right)⟹B^{\top }\in \left(V^{\ast \ast }\stackrel{\text{ linear}}{\to }{V}^{\ast }\right)$$

where $V^{\ast \ast }$ is simply $V$. So, $B,B^{\top }$ are of the same type.

Definition (Symmetric Bilinear Forms)

A bilinear form $B$ is symmetric if $B=B^{\top }$. This is the same as $B(u)(v)=B(v)(u)$ or $B_{ij}=B_{ji}$.

Theorem (Symmetric Bilinear Forms and Quadratic Forms)

There is a one-to-one correspondence between symmetric bilinear forms and quadratic forms.

• From symmetric to quadratic: $q(u)=B(u)(u)$.
• From quadratic to symmetric: $q(u+v)=q(u)+q(v)+2B(u)(v)$

Definition (Positive Definite)

A symmetric bilinear form is positive definite if its quadratic form takes positive values $q(u)>0$ for all $u\ne 0$.

Definition (Inner Product Structure or Metric)

An inner product structure or metric on $V$ is a positive definite symmetric bilinear form. It is often denoted by $g_{ij}$ or ${♭}_{ij}$, the flat operator.

$$g(u,v)=⟨u,v⟩=⟨♭u|v⟩$$

There are infinitely many choices for a metric on $V$. That is, for a plain vector space, there is no canonical choice of metric. The metric (as any bilinear form) is a map from vectors to covectors, so the flat operator is known as index-lowering operator.

$$(♭u{)}_i={♭}_{ij}{u}^j$$

• The flat operator is invertible (due to positive-definiteness).
• The inverse of the flat operator is known as the index-raising operator or sharp operator $♯={♭}^{-1}$.

$${\left(♯\alpha \right)}^i={♯}^{ij}{\alpha }_j$$

Via index notation, the label is $g^{ij}=(g^{-1}{)}^{ij}$.

In particular, an inner product structure $♭:V\to V^{\ast }$ can turn vectors to covectors and vice versa with the inverse $♯:V^{\ast }\to V$. Geometrically, we can draw a quadratic surface with some vector $\vec{v}$. The covector is then the hyperplane ${\vec{v}}^{♭}$ where $♭(\vec{v})(\vec{v})=1$ denotes the unit sphere. The vectors are the tangent vectors to the surface. We construct the polar plane across the quadric surface.