Unit 8: Canonical Forms, Bilinear, Quadratic & Hermitian Forms

1. Schur's Triangularization Theorem

Not all square matrices are diagonalizable. However, every square complex matrix can be triangularized by a unitary transformation.

Theorem 8.1 (Schur's Triangularization Theorem):

Let $A \in M_n(\mathbb{C})$. There exists a unitary matrix $U \in U(n)$ ($U^\dagger U = I_n$) such that:

where $T$ is an upper triangular matrix:

The diagonal entries $\lambda_1, \dots, \lambda_n$ of $T$ are precisely the eigenvalues of $A$ (counted with algebraic multiplicities).

Proof (by Induction on $n$):

• Base Case: For $n = 1$, $A$ is a $1 \times 1$ scalar; $U = (1)$ is unitary and $T = A$ is triangular.
• Inductive Hypothesis: Assume every $(n-1) \times (n-1)$ matrix is unitarily triangularizable.
• Inductive Step:

By the Fundamental Theorem of Algebra, $A$ has at least one eigenvalue $\lambda_1 \in \mathbb{C}$ with unit eigenvector $\vec{u}_1$ ($\|\vec{u}_1\| = 1$). Using Gram-Schmidt, extend $\{\vec{u}_1\}$ to an orthonormal basis $\{\vec{u}_1, \vec{w}_2, \dots, \vec{w}_n\}$ of $\mathbb{C}^n$. Form the unitary matrix $U_1 = \begin{pmatrix} \vec{u}_1 & \vec{w}_2 & \cdots & \vec{w}_n \end{pmatrix}$. Then:

where $A_1 \in M_{n-1}(\mathbb{C})$. By induction, there exists unitary $U_2 \in U(n-1)$ such that $U_2^\dagger A_1 U_2 = T_1$ is upper triangular. Setting $U = U_1 \begin{pmatrix} 1 & \vec{0}^T \\ \vec{0} & U_2 \end{pmatrix}$, we find $U^\dagger A U = \begin{pmatrix} \lambda_1 & \vec{r}^T U_2 \\ \vec{0} & T_1 \end{pmatrix} = T$ is upper triangular. $\blacksquare$

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2. Generalized Eigenvectors and Eigenspaces

When a matrix $A$ has an eigenvalue $\lambda$ whose geometric multiplicity $g_\lambda = \dim \ker(A - \lambda I)$ is strictly less than its algebraic multiplicity $a_\lambda$, the eigenspace $E_\lambda$ is insufficient to span $\mathbb{C}^{a_\lambda}$. We must seek generalized eigenvectors.

Definition:

A non-zero vector $\vec{v} \in V$ is a Generalized Eigenvector of rank $k$ corresponding to eigenvalue $\lambda$ if:

Generalized Eigenspace:

The Generalized Eigenspace corresponding to eigenvalue $\lambda$ is:

Fundamental Properties:

1. $E_\lambda \subseteq K_\lambda$.
2. $\dim K_\lambda = a_\lambda$ (the algebraic multiplicity of $\lambda$).
3. $K_\lambda$ is an $A$-invariant subspace: $A(K_\lambda) \subseteq K_\lambda$.

Theorem 8.2 (Primary Decomposition Theorem):

Let $T: V \to V$ be a linear operator on a finite-dimensional complex vector space $V$, with distinct eigenvalues $\lambda_1, \dots, \lambda_k$. Then $V$ decomposes into the direct sum of its generalized eigenspaces:

Each $K_{\lambda_i}$ is $T$-invariant, and $(T - \lambda_i I)$ restricted to $K_{\lambda_i}$ is nilpotent.

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3. Nilpotent Operators and Jordan Chains

An operator $N: W \to W$ is nilpotent if $N^p = O$ for some positive integer $p$. The smallest such $p$ is the index of nilpotency of $N$.

Jordan Chains (Cyclic Subspaces for Nilpotent Operators):

Let $\vec{v}$ be a vector such that $N^{k-1} \vec{v} \neq \vec{0}$ but $N^k \vec{v} = \vec{0}$. The sequence of $k$ vectors:

is linearly independent and spans an $N$-invariant subspace of dimension $k$. With respect to this ordered basis, the matrix of $N$ is the standard nilpotent Jordan block:

### 1. The Jordan Canonical Form (JCF) #### Definition of a Jordan Block: A **Jordan Block** of size $k$ corresponding to eigenvalue $\lambda$, denoted $J_k(\lambda)$, is a $k \times k$ upper bidiagonal matrix: $$J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}_{k \times k} = \lambda I_k + N_k$$ #### Theorem 8.3 (Jordan Canonical Form Theorem): Let $A \in M_n(\mathbb{C})$. There exists an invertible matrix $P \in GL_n(\mathbb{C})$ such that: $$P^{-1} A P = J = \begin{pmatrix} J_{k_1}(\lambda_1) & O & \cdots & O \\ O & J_{k_2}(\lambda_2) & \cdots & O \\ \vdots & \vdots & \ddots & \vdots \\ O & O & \cdots & J_{k_m}(\lambda_m) \end{pmatrix}$$ where each $J_{k_i}(\lambda_i)$ is a Jordan block. The matrix $J$ is unique up to the ordering of the blocks. --- --- ### 2. Determining Jordan Block Structure from Ranks Let $\lambda$ be an eigenvalue of $A$, and define $r_k = \text{rank}(A - \lambda I)^k$ (with $r_0 = n$). - **Total number of Jordan blocks for $\lambda$:** $$\text{Number of blocks} = \dim \ker(A - \lambda I) = n - r_1 = g_\lambda \quad (\text{geometric multiplicity})$$ - **Number of Jordan blocks of size $\ge k$:** $$N_{\ge k} = r_{k-1} - r_k$$ - **Number of Jordan blocks of size exactly $k$:** $$N_k = N_{\ge k} - N_{\ge k+1} = (r_{k-1} - r_k) - (r_k - r_{k+1}) = r_{k-1} - 2r_k + r_{k+1}$$ #### Relation to the Minimal Polynomial: The size of the **largest** Jordan block associated with eigenvalue $\lambda_i$ equals the multiplicity of $(x - \lambda_i)$ in the **minimal polynomial** $m_A(x)$. That is, if: $$p_A(x) = \prod_{i=1}^d (x - \lambda_i)^{a_i} \quad \text{and} \quad m_A(x) = \prod_{i=1}^d (x - \lambda_i)^{m_i}$$ then: - $\sum (\text{sizes of all blocks for } \lambda_i) = a_i$ - $\max (\text{size of a block for } \lambda_i) = m_i$ --- ### 3. The Rational Canonical Form (Frobenius Normal Form) Unlike the Jordan form, which requires the scalar field to contain all eigenvalues (algebraically closed field like $\mathbb{C}$), the **Rational Canonical Form** is valid over **any** field $F$ (such as $\mathbb{Q}$ or $\mathbb{R}$). #### Companion Matrix: For a monic polynomial $p(x) = x^k + c_{k-1} x^{k-1} + \cdots + c_1 x + c_0$, its **Companion Matrix** is: $$C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -c_{k-1} \end{pmatrix}$$ The characteristic polynomial and minimal polynomial of $C(p)$ are both equal to $p(x)$. #### Invariant Factor Decomposition: Every matrix $A \in M_n(F)$ is similar over $F$ to a unique block-diagonal matrix: $$R = \text{diag}(C(d_1(x)), \, C(d_2(x)), \, \dots, \, C(d_m(x)))$$ where the monic polynomials $d_1(x), \dots, d_m(x)$ are the **Invariant Factors** of $A$, satisfying: $$d_1(x) \mid d_2(x) \mid \cdots \mid d_m(x)$$ Moreover: - $d_m(x) = m_A(x)$ (the minimal polynomial of $A$) - $d_1(x) d_2(x) \cdots d_m(x) = p_A(x)$ (the characteristic polynomial of $A$)

1. Bilinear Forms

Let $V$ be a vector space over field $F$. A function $B: V \times V \to F$ is a Bilinear Form if it is linear in each argument separately:

1. $B(c_1 \vec{u}_1 + c_2 \vec{u}_2, \, \vec{v}) = c_1 B(\vec{u}_1, \vec{v}) + c_2 B(\vec{u}_2, \vec{v})$
2. $B(\vec{u}, \, c_1 \vec{v}_1 + c_2 \vec{v}_2) = c_1 B(\vec{u}, \vec{v}_1) + c_2 B(\vec{u}, \vec{v}_2)$

Matrix Representation:

Let $\beta = \{\vec{e}_1, \dots, \vec{e}_n\}$ be an ordered basis of $V$. The matrix of $B$ relative to $\beta$ is $[B]_\beta = (b_{ij}) \in M_n(F)$, where:

For any vectors $\vec{x} = \sum x_i \vec{e}_i$ and $\vec{y} = \sum y_j \vec{e}_j$:

Congruence Transformation under Change of Basis:

If $\beta'$ is another basis of $V$ with transition matrix $P$ ($[\vec{x}]_\beta = P [\vec{x}]_{\beta'}$), then:

Thus the matrix transforms by Congruence:

Two matrices $A, B$ are congruent if there exists an invertible matrix $P$ such that $B = P^T A P$.

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2. Quadratic Forms

Let $V$ be a vector space over $\mathbb{R}$. A function $q: V \to \mathbb{R}$ is a Quadratic Form if $q(\vec{x}) = B(\vec{x}, \vec{x})$ for some symmetric bilinear form $B$. In coordinates:

where $A = A^T = \frac{1}{2}(M + M^T)$ is the unique symmetric matrix associated with $q$. Expanding explicitly:

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3. Diagonalization of Quadratic Forms

Every quadratic form can be reduced to a diagonal form (sum and difference of squares) by a non-singular linear change of variables $\vec{x} = P \vec{y}$:

Method 1: Lagrange's Method of Completing Squares

An algebraic, non-orthogonal procedure:

• Case 1 (At least one diagonal coefficient $a_{ii} \neq 0$):

Suppose $a_{11} \neq 0$. Group all terms containing $x_1$:

Complete the square:

Set $y_1 = x_1 + \sum_{j=2}^n \frac{a_{1j}}{a_{11}} x_j$, and repeat inductively on $q'(x_2, \dots, x_n)$.

• Case 2 (All $a_{ii} = 0$, but some cross term $a_{ij} \neq 0$):

Make the non-singular substitution $x_i = u_i + u_j$, $x_j = u_i - u_j$, so $2 a_{ij} x_i x_j = 2 a_{ij}(u_i^2 - u_j^2)$, producing non-zero diagonal coefficients, and revert to Case 1.

Method 2: Orthogonal Diagonalization (Principal Axis Theorem)

By the Real Spectral Theorem, $A = Q \Lambda Q^T$ with $Q \in O(n)$ orthogonal ($Q^T = Q^{-1}$). Letting $\vec{x} = Q \vec{y}$ gives:

This rotation eliminates cross-terms while geometrically preserving angles and distances!

1. Sylvester's Law of Inertia

Let $q$ be a real quadratic form on an $n$-dimensional real vector space $V$. Through an invertible change of variables $\vec{x} = P \vec{y}$, $q$ can always be brought into its Canonical Form:

where:

• $p$ is the number of positive squares (positive index of inertia).
• $q$ is the number of negative squares (negative index of inertia).
• $r = p + q$ is the rank of the quadratic form ($r = \text{rank}(A)$).
• $z = n - r$ is the nullity (number of missing variables).
• $s = p - q$ is the signature of the form.

Theorem 8.4 (Sylvester's Law of Inertia):

The integers $p$ and $q$ are independent of the choice of diagonalizing coordinate transformation. They depend solely on the quadratic form $q$ itself.

Proof: Suppose there exist two non-singular changes of coordinates reducing $q$ to:

Assume, for contradiction, that $p > p'$. Let $\beta = \{\vec{u}_1, \dots, \vec{u}_n\}$ be the basis in which $q$ has coordinates $\vec{y}$, and $\gamma = \{\vec{w}_1, \dots, \vec{w}_n\}$ the basis for $\vec{z}$. Define two subspaces of $V$:

For any non-zero $\vec{v} \in V^+$: $q(\vec{v}) = y_1^2 + \cdots + y_p^2 > 0$. For any $\vec{v} \in V^-$: $q(\vec{v}) = -z_{p'+1}^2 - \cdots - z_{p'+q'}^2 \le 0$. Hence $V^+ \cap V^- = \{\vec{0}\}$. By the dimension formula for subspace sums:

Since $p > p'$, $n + (p - p') > n = \dim V$, which contradicts the fact that $V^+ + V^- \subseteq V$ cannot have dimension strictly exceeding $n$. Therefore $p \le p'$. By symmetry, $p' \le p \implies p = p'$. Since the rank $r = p + q = p' + q'$ is invariant (it is the rank of the matrix $A$), it follows that $q = q'$. $\blacksquare$

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2. Definiteness Classification of Quadratic Forms

A real quadratic form $q(\vec{x}) = \vec{x}^T A \vec{x}$ is classified as:

1. Positive Definite: $q(\vec{x}) > 0$ for all $\vec{x} \neq \vec{0}$.

• Spectrum: All eigenvalues $\lambda_i > 0$.
• Inertia: $p = n, q = 0$.

2. Positive Semidefinite: $q(\vec{x}) \ge 0$ for all $\vec{x} \in \mathbb{R}^n$.

• Spectrum: All eigenvalues $\lambda_i \ge 0$.
• Inertia: $p \le n, q = 0$.

3. Negative Definite: $q(\vec{x}) < 0$ for all $\vec{x} \neq \vec{0}$.

• Spectrum: All eigenvalues $\lambda_i < 0$.
• Inertia: $p = 0, q = n$.

4. Negative Semidefinite: $q(\vec{x}) \le 0$ for all $\vec{x} \in \mathbb{R}^n$.

• Spectrum: All eigenvalues $\lambda_i \le 0$.
• Inertia: $p = 0, q \le n$.

5. Indefinite: $q(\vec{x})$ takes both strictly positive and strictly negative values.

• Spectrum: There exists at least one $\lambda_i > 0$ and at least one $\lambda_j < 0$.
• Inertia: $p > 0$ and $q > 0$.

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3. Sylvester's Leading Principal Minors Criterion

The Leading Principal Submatrices of $A = (a_{ij}) \in M_n(\mathbb{R})$ are:

Their determinants $\Delta_k = \det(A_k)$ are called the Leading Principal Minors.

Theorem 8.5 (Sylvester's Criterion):

1. $A$ is Positive Definite if and only if all leading principal minors are strictly positive:

1. $A$ is Negative Definite if and only if the leading principal minors alternate in sign, starting negative:

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4. Hermitian Forms

Over the complex field $\mathbb{C}$, the analogue of a symmetric bilinear form is a Hermitian Form $H: V \times V \to \mathbb{C}$:

1. $H(c_1 \vec{u}_1 + c_2 \vec{u}_2, \, \vec{v}) = c_1 H(\vec{u}_1, \vec{v}) + c_2 H(\vec{u}_2, \vec{v})$
2. $H(\vec{v}, \vec{u}) = \overline{H(\vec{u}, \vec{v})}$

The associated complex quadratic form is:

where $A^\dagger = A$ is a Hermitian matrix. Crucially, $q(\vec{x})$ is always real-valued:

Under a non-singular complex change of basis $\vec{x} = P \vec{y}$, the matrix transforms by Hermitian Congruence:

Sylvester's Law of Inertia holds identically for Hermitian forms: every Hermitian matrix is congruent to a diagonal matrix with entries $\{+1, -1, 0\}$, with the counts of $+1$ and $-1$ being invariant.