Multilinear Algebra and Determinants

9A. Bilinear Forms and Quadratic Forms

1) Bilinear Form

Bilinear Form

A bilinear form on $V$ is a function

$$\beta:V\times V\to\mathbb{F}$$

that is linear in each argument separately.

Meaning: A bilinear form takes two vectors and produces a scalar, behaving linearly in either vector when the other is fixed.

2) Space of Bilinear Forms

$V^{(2)}$

The set of all bilinear forms on $V$ is denoted by

$$V^{(2)}.$$

It is a vector space.

Meaning: Bilinear forms themselves can be added and multiplied by scalars.

3) Matrix of a Bilinear Form

Matrix of a Bilinear Form

Suppose

$$e_1,\ldots,e_n$$

is a basis of $V$.

The matrix of a bilinear form $\beta$ is defined by

$$\mathcal{M}(\beta)_{j,k} = \beta(e_j,e_k).$$

Meaning: A bilinear form is completely represented by its values on pairs of basis vectors.

Dimension

$$\dim V^{(2)} = (\dim V)^2.$$

Meaning: An $n$-dimensional vector space requires $n^2$ scalar values to specify a bilinear form.

4) Change of Basis

Change-of-Basis Formula

If $A$ and $B$ represent the same bilinear form in two bases and $C$ is the corresponding change-of-basis matrix, then

$$A=C^tBC.$$

Meaning: Bilinear-form matrices change by multiplication with the transpose of the basis-change matrix on one side.

5) Symmetric Bilinear Form

Symmetric Bilinear Form

A bilinear form $\rho$ is symmetric if

$$\rho(u,w)=\rho(w,u)$$

for all $u,w\in V$.

Meaning: Swapping the two input vectors does not change the value.

Symmetric Matrix

A square matrix $A$ is symmetric if

$$A=A^t.$$

Meaning: A symmetric matrix is unchanged by transposition.

6) Symmetric Bilinear Forms and Matrices

Symmetry Criterion

A bilinear form $\rho$ is symmetric if and only if its matrix is symmetric with respect to any basis.

Meaning: Symmetry is a property of the bilinear form itself, not of a particular basis.

7) Diagonalization of a Symmetric Bilinear Form

Diagonalization

Every symmetric bilinear form has a diagonal matrix with respect to some basis.

Meaning: A suitable basis removes all cross terms from a symmetric bilinear form.

Real Inner Product Case

If $V$ is a real inner product space, the basis can be chosen to be orthonormal.

Meaning: Over a real inner product space, a symmetric bilinear form can be diagonalized without losing orthonormality.

8) Alternating Bilinear Form

Alternating Bilinear Form

A bilinear form $\alpha$ is alternating if

$$\alpha(v,v)=0$$

for every $v\in V$.

Meaning: An alternating bilinear form vanishes whenever its two inputs are equal.

Equivalent Condition

$$\alpha(u,w) = -\alpha(w,u).$$

Meaning: Swapping the two vectors reverses the sign.

9) Decomposition of Bilinear Forms

Symmetric-Alternating Decomposition

Every bilinear form can be uniquely written as the sum of a symmetric and an alternating bilinear form.

$$V^{(2)} = V_{\mathrm{sym}}^{(2)} \oplus V_{\mathrm{alt}}^{(2)}.$$

For $\beta\in V^{(2)}$,

$$\rho(u,w) = \frac{\beta(u,w)+\beta(w,u)}{2},$$

and

$$\alpha(u,w) = \frac{\beta(u,w)-\beta(w,u)}{2}.$$

Meaning: Every bilinear form splits uniquely into a symmetric part and an antisymmetric part.

10) Quadratic Form

Quadratic Form

For a bilinear form $\beta$, define

$$q_\beta(v) = \beta(v,v).$$

A function

$$q:V\to\mathbb{F}$$

is a quadratic form if

$$q=q_\beta$$

for some bilinear form $\beta$.

Meaning: A quadratic form is obtained by applying a bilinear form to the same vector twice.

11) Symmetric Bilinear Form Associated with a Quadratic Form

Unique Symmetric Bilinear Form

Every quadratic form $q$ has a unique symmetric bilinear form $\rho$ such that

$$q(v)=\rho(v,v).$$

It can be recovered from $q$ by

$$\rho(u,w) = \frac{ q(u+w)-q(u)-q(w) }{2}.$$

Meaning: A quadratic form contains exactly the information of one symmetric bilinear form.

12) Homogeneity of a Quadratic Form

Quadratic Scaling

For every $\lambda\in\mathbb{F}$,

$$q(\lambda v) = \lambda^2q(v).$$

Meaning: Scaling a vector by $\lambda$ scales its quadratic value by $\lambda^2$.

13) Diagonalization of a Quadratic Form

Diagonal Form

For every quadratic form $q$, there exist a basis

$$e_1,\ldots,e_n$$

and scalars

$$\lambda_1,\ldots,\lambda_n$$

such that

$$q(x_1e_1+\cdots+x_ne_n) = \lambda_1x_1^2+\cdots+\lambda_nx_n^2.$$

Meaning: Every quadratic form can be expressed without cross terms using a suitable basis.

Real Inner Product Case

If $V$ is a real inner product space, the basis can be chosen to be orthonormal.

Meaning: Real quadratic forms can be diagonalized using perpendicular unit directions.

9B. Alternating Multilinear Forms

1) Multilinear Form

$m$-Linear Form

An $m$-linear form is a function

$$\beta:V^m\to\mathbb{F}$$

that is linear in each argument when all other arguments are fixed.

Meaning: A multilinear form generalizes a bilinear form from two inputs to any finite number of inputs.

$V^{(m)}$

The vector space of all $m$-linear forms on $V$ is denoted by

$$V^{(m)}.$$

Meaning: $V^{(m)}$ contains all scalar-valued functions that are linear in each of $m$ vector arguments.

2) Alternating Multilinear Form

Alternating $m$-Linear Form

An $m$-linear form $\alpha$ is alternating if

$$\alpha(v_1,\ldots,v_m)=0$$

whenever two of the input vectors are equal.

Meaning: An alternating form vanishes when two input directions are repeated.

3) Alternating Forms and Linear Dependence

Linear Dependence Property

If

$$v_1,\ldots,v_m$$

is linearly dependent and $\alpha$ is alternating, then

$$\alpha(v_1,\ldots,v_m)=0.$$

Meaning: An alternating form can produce a nonzero value only from sufficiently independent input directions.

4) Too Many Inputs

Dimension Restriction

If

$$m>\dim V,$$

then the only alternating $m$-linear form on $V$ is the zero form.

Meaning: More vectors than the dimension of the space must be linearly dependent.

5) Swapping Inputs

Sign Change

Swapping any two inputs of an alternating multilinear form changes its sign.

$$\alpha(\ldots,v_j,\ldots,v_k,\ldots) = -\alpha(\ldots,v_k,\ldots,v_j,\ldots).$$

Meaning: Each swap contributes a factor of $-1$.

6) Permutation

Permutation

A permutation of

$$(1,\ldots,m)$$

is a rearrangement containing every number $1,\ldots,m$ exactly once.

Meaning: A permutation records a possible reordering of the input positions.

7) Sign of a Permutation

Sign

The sign of a permutation is

$$\operatorname{sign}(j_1,\ldots,j_m) = (-1)^N,$$

where $N$ is the number of inversions.

Meaning: The sign is $1$ for an even permutation and $-1$ for an odd permutation.

8) Permutations and Alternating Forms

Permutation Formula

For an alternating $m$-linear form,

$$\alpha(v_{j_1},\ldots,v_{j_m}) = \operatorname{sign}(j_1,\ldots,j_m) \alpha(v_1,\ldots,v_m).$$

Meaning: Reordering the vectors changes only the sign according to the permutation.

9) Top-Degree Alternating Forms

One-Dimensionality

If

$$n=\dim V,$$

then

$$\dim V_{\mathrm{alt}}^{(n)}=1.$$

Meaning: Up to multiplication by a scalar, there is only one alternating $n$-linear form on an $n$-dimensional space.

10) Alternating Forms and Bases

Linear Independence Criterion

Let

$$n=\dim V$$

and let $\alpha$ be a nonzero alternating $n$-linear form.

Then

$$\alpha(e_1,\ldots,e_n)\neq0$$

if and only if

$$e_1,\ldots,e_n$$

is linearly independent.

Meaning: A top-degree alternating form detects whether $n$ vectors form a basis.

9C. Determinants

1) Determinant of an Operator

Determinant

Let

$$n=\dim V$$

and $T\in\mathcal{L}(V)$.

The determinant of $T$ is the unique scalar $\det T$ such that

$$\alpha(Tv_1,\ldots,Tv_n) = (\det T)\alpha(v_1,\ldots,v_n)$$

for every alternating $n$-linear form $\alpha$.

Meaning: The determinant measures how $T$ scales an alternating top-dimensional quantity.

2) Basic Determinants

Identity

$$\det I=1.$$

Scalar Multiple of the Identity

$$\det(\lambda I) = \lambda^n.$$

Scalar Multiple of an Operator

$$\det(\lambda T) = \lambda^n\det T.$$

Meaning: Determinants scale according to the dimension of the vector space.

3) Determinant of a Matrix

Matrix Determinant

For a square matrix $A$, define

$$\det A$$

to be the determinant of the corresponding operator on $\mathbb{F}^n$.

Meaning: The usual matrix determinant is the coordinate representation of the determinant of an operator.

4) Determinant as an Alternating Multilinear Form

Column Interpretation

For an $n$-by-$n$ matrix with columns

$$v_1,\ldots,v_n,$$

the function

$$(v_1,\ldots,v_n) \mapsto \det(v_1\ \cdots\ v_n)$$

is an alternating $n$-linear form.

Meaning: The determinant is linear in each column and changes sign when two columns are swapped.

5) Determinant Formula

Permutation Formula

For an $n$-by-$n$ matrix $A$,

$$\det A = \sum_{\sigma} \operatorname{sign}(\sigma) \prod_{k=1}^{n} A_{\sigma(k),k}.$$

Meaning: The determinant combines one entry from every row and column, with a sign determined by the permutation.

6) Upper-Triangular Matrix

Triangular Determinant

If $A$ is upper triangular with diagonal entries

$$\lambda_1,\ldots,\lambda_n,$$

then

$$\det A = \lambda_1\cdots\lambda_n.$$

Meaning: The determinant of a triangular matrix is simply the product of its diagonal entries.

7) Multiplicativity

Multiplicative Property

For operators,

$$\det(ST) = (\det S)(\det T).$$

For square matrices,

$$\det(AB) = (\det A)(\det B).$$

Meaning: Volume-scaling factors multiply when transformations are composed.

8) Determinant and Invertibility

Invertibility Criterion

$$T\text{ is invertible} \iff \det T\neq0.$$

Similarly,

$$A\text{ is invertible} \iff \det A\neq0.$$

Meaning: A zero determinant means that some dimension has been collapsed.

Inverse

If $T$ is invertible, then

$$\det(T^{-1}) = \frac{1}{\det T}.$$

Meaning: The inverse reverses the determinant scaling.

9) Determinant and Eigenvalues

Eigenvalue Criterion

A scalar $\lambda$ is an eigenvalue of $T$ if and only if

$$\det(\lambda I-T)=0.$$

Meaning: Eigenvalues are precisely the values that make $\lambda I-T$ noninvertible.

10) Similarity Invariance

Similarity

If $S$ is invertible, then

$$\det(S^{-1}TS) = \det T.$$

Meaning: Changing coordinates does not change the determinant of an operator.

11) Operator and Matrix Determinants

Basis Independence

For any basis of $V$,

$$\det T = \det\mathcal{M}(T).$$

Meaning: The determinant is independent of which matrix representation of the operator is used.

12) Determinant and Eigenvalue Product

Product of Eigenvalues

If

$$\mathbb{F}=\mathbb{C},$$

then

$$\det T$$

equals the product of all eigenvalues of $T$, counted according to their multiplicities.

$$\det T = \lambda_1\cdots\lambda_n.$$

Meaning: The determinant is the total product of the eigenvalue scaling factors.

13) Transpose, Dual, and Adjoint

Transpose

$$\det A^t = \det A.$$

Meaning: Transposing a matrix does not change its determinant.

Dual

$$\det T' = \det T.$$

Meaning: An operator and its dual have the same determinant.

Adjoint

For an operator on an inner product space,

$$\det(T^*) = \overline{\det T}.$$

Meaning: Taking the adjoint conjugates the determinant.

14) Row and Column Operations

Equal Rows or Columns

If two rows or two columns are equal, then

$$\det A=0.$$

Meaning: Repeated rows or columns indicate linear dependence.

Swap

Swapping two rows or two columns multiplies the determinant by

$$-1.$$

Meaning: Exchanging two directions reverses orientation.

Scaling

Multiplying one row or column by $\lambda$ multiplies the determinant by $\lambda$.

Meaning: Scaling one direction scales the determinant by the same amount.

Row or Column Addition

Adding a scalar multiple of one row to another row does not change the determinant.

The same holds for columns.

Meaning: Shearing does not change the determinant.

15) Determinant of a Unitary Operator

Unitary Determinant

If $S$ is unitary, then

$$|\det S|=1.$$

Meaning: A unitary transformation preserves volume.

16) Determinant of a Positive Operator

Positive Determinant

If $T$ is positive, then

$$\det T\ge0.$$

Meaning: A positive operator has only nonnegative eigenvalue scaling factors.

17) Determinant and Singular Values

Singular Value Product

If

$$s_1,\ldots,s_n$$

are the singular values of $T$, then

$$|\det T| = s_1\cdots s_n.$$

Equivalently,

$$|\det T| = \sqrt{\det(T^*T)}.$$

Meaning: The absolute determinant is the product of all orthogonal stretching factors.

18) Geometric Meaning of the Determinant

Volume Scaling

For $T\in\mathcal{L}(\mathbb{R}^n)$,

$$\operatorname{volume}(T(\Omega)) = |\det T| \operatorname{volume}(\Omega).$$

Meaning: The absolute determinant tells how much a linear transformation scales $n$-dimensional volume.

19) Characteristic Polynomial

Characteristic Polynomial

For any finite-dimensional real or complex vector space,

$$p_T(z) = \det(zI-T).$$

Meaning: The characteristic polynomial records the values for which $zI-T$ becomes noninvertible.

Degree

$$\deg p_T = \dim V.$$

Meaning: The characteristic polynomial of an $n$-dimensional operator has degree $n$.

Eigenvalues

$$p_T(\lambda)=0$$

if and only if $\lambda$ is an eigenvalue in $\mathbb{F}$.

Meaning: Eigenvalues are the roots of the characteristic polynomial.

20) Cayley–Hamilton Theorem

Cayley–Hamilton Theorem

Every operator satisfies its own characteristic polynomial.

$$p_T(T)=0.$$

Meaning: Substituting the operator itself into its characteristic polynomial gives the zero operator.

21) Minimal and Characteristic Polynomials

Divisibility

The minimal polynomial divides the characteristic polynomial.

$$p_{\min}\mid p_T.$$

Meaning: The characteristic polynomial always contains enough factors to annihilate the operator.

22) Trace and Determinant in the Characteristic Polynomial

Characteristic Polynomial Coefficients

If

$$n=\dim V,$$

then

$$p_T(z) = z^n - (\operatorname{tr}T)z^{n-1} +\cdots+ (-1)^n\det T.$$

Meaning: Trace and determinant appear directly as important coefficients of the characteristic polynomial.

23) Hadamard's Inequality

Hadamard's Inequality

If

$$v_1,\ldots,v_n$$

are the columns of $A$, then

$$|\det A| \le \prod_{k=1}^{n}\|v_k\|.$$

Meaning: For fixed column lengths, the largest possible volume occurs when the columns are orthogonal.

9D. Tensor Products

1) Bilinear Functional

Bilinear Functional

A bilinear functional on $V\times W$ is a function

$$\beta:V\times W\to\mathbb{F}$$

that is linear in each argument separately.

Meaning: A bilinear functional combines vectors from two possibly different vector spaces into a scalar.

$\mathcal{B}(V,W)$

The space of all bilinear functionals on $V\times W$ is denoted by

$$\mathcal{B}(V,W).$$

Its dimension is

$$\dim\mathcal{B}(V,W) = (\dim V)(\dim W).$$

Meaning: A bilinear functional requires one coefficient for every pair of basis directions.

2) Tensor Product of Vector Spaces

Tensor Product

The tensor product of $V$ and $W$ is defined by

$$V\otimes W = \mathcal{B}(V',W').$$

Meaning: $V\otimes W$ is a new vector space designed to represent bilinear combinations of vectors from $V$ and $W$.

3) Tensor Product of Vectors

Simple Tensor

For

$$v\in V, \qquad w\in W,$$

the tensor

$$v\otimes w$$

is defined by

$$(v\otimes w)(\varphi,\tau) = \varphi(v)\tau(w)$$

for $\varphi\in V'$ and $\tau\in W'$.

Meaning: $v\otimes w$ combines one vector from each space into an element of the tensor-product space.

4) Dimension of a Tensor Product

Tensor Product Dimension

$$\dim(V\otimes W) = (\dim V)(\dim W).$$

Meaning: Tensor-product dimensions multiply rather than add.

5) Bilinearity of the Tensor Product

Addition in the First Factor

$$(v_1+v_2)\otimes w = v_1\otimes w+v_2\otimes w.$$

Addition in the Second Factor

$$v\otimes(w_1+w_2) = v\otimes w_1+v\otimes w_2.$$

Scalar Multiplication

$$\lambda(v\otimes w) = (\lambda v)\otimes w = v\otimes(\lambda w).$$

Meaning: Tensor multiplication is linear in each vector separately.

6) Basis of a Tensor Product

Tensor Product Basis

If

$$e_1,\ldots,e_m$$

is a basis of $V$ and

$$f_1,\ldots,f_n$$

is a basis of $W$, then

$$\{e_j\otimes f_k\}$$

is a basis of

$$V\otimes W.$$

Meaning: Every pair of basis directions produces one independent tensor-product direction.

7) Simple Tensors and General Tensors

General Tensor

Every element of $V\otimes W$ can be written as a finite sum

$$\sum_k v_k\otimes w_k.$$

However, not every tensor must itself have the form

$$v\otimes w.$$

Meaning: A single tensor product $v\otimes w$ is only a special type of element in the full tensor-product space.

8) Bilinear Map

Bilinear Map

A bilinear map is a function

$$\Gamma:V\times W\to U$$

that is linear in each argument separately.

Meaning: Unlike a bilinear functional, a bilinear map may output vectors instead of only scalars.

9) Universal Property of Tensor Products

Bilinear Maps Become Linear Maps

For every bilinear map

$$\Gamma:V\times W\to U,$$

there exists a unique linear map

$$\widehat{\Gamma}:V\otimes W\to U$$

such that

$$\widehat{\Gamma}(v\otimes w) = \Gamma(v,w).$$

Meaning: Tensor products convert bilinear problems into ordinary linear problems.

10) Tensor Product of Inner Product Spaces

Tensor Product Inner Product

If $V$ and $W$ are inner product spaces, there is a unique inner product on $V\otimes W$ satisfying

$$\langle v\otimes w, u\otimes x \rangle = \langle v,u\rangle \langle w,x\rangle.$$

Meaning: Inner products of simple tensors are products of the corresponding inner products.

Orthonormal Tensor Basis

If

$$e_1,\ldots,e_m$$

and

$$f_1,\ldots,f_n$$

are orthonormal bases, then

$$\{e_j\otimes f_k\}$$

is an orthonormal basis of $V\otimes W$.

Meaning: Tensoring orthonormal bases produces an orthonormal basis of the tensor-product space.

11) Tensor Products of Multiple Vector Spaces

Multiple Tensor Product

For vector spaces

$$V_1,\ldots,V_m,$$

the tensor product is denoted by

$$V_1\otimes\cdots\otimes V_m.$$

Meaning: Tensor products extend naturally from two vector spaces to any finite number of vector spaces.

12) Tensor Product of Multiple Vectors

Multiple Simple Tensor

For

$$v_k\in V_k,$$

write

$$v_1\otimes\cdots\otimes v_m.$$

It satisfies multilinearity in all $m$ arguments.

Meaning: The tensor product is linear independently in every factor.

13) Dimension of a Multiple Tensor Product

Dimension

$$\dim(V_1\otimes\cdots\otimes V_m) = \prod_{k=1}^{m}\dim V_k.$$

Meaning: The dimensions of all tensor factors multiply.

14) Basis of a Multiple Tensor Product

Tensor Basis

If

$$e_1^k,\ldots,e_{n_k}^k$$

is a basis of $V_k$ for every $k$, then all tensors

$$e_{j_1}^1 \otimes\cdots\otimes e_{j_m}^m$$

form a basis of

$$V_1\otimes\cdots\otimes V_m.$$

Meaning: Choosing one basis vector from each factor produces every basis direction of the tensor product.

15) Universal Property for Multiple Tensor Products

Multilinear Maps Become Linear Maps

Every $m$-linear map

$$\Gamma: V_1\times\cdots\times V_m \to U$$

corresponds to a unique linear map

$$\widehat{\Gamma}: V_1\otimes\cdots\otimes V_m \to U$$

such that

$$\widehat{\Gamma} (v_1\otimes\cdots\otimes v_m) = \Gamma(v_1,\ldots,v_m).$$

Meaning: Tensor products convert multilinear maps into ordinary linear maps.