A bilinear form is linear one slot at a time

Key idea

For vector spaces over the same field, a bilinear form is a scalar-valued map \(B:V\times V\to F\) such that \(B(ax_1+bx_2,y)=aB(x_1,y)+bB(x_2,y)\) and \(B(x,ay_1+by_2)=aB(x,y_1)+bB(x,y_2)\). In coordinates, every bilinear form on \(F^n\) has the shape \(B(x,y)=x^TAy\).

Recognition checklist

• No powers like \(x_1^2\) inside the first slot test.
• No lengths such as \(\|y\|\), because norms are not linear.
• Products \(x_i y_j\) are acceptable: each slot still appears linearly when the other is fixed.
• Symmetric is a separate property; a bilinear form can be bilinear without being symmetric.

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Quadratic forms come from evaluating a symmetric form twice

Key idea

If \(B\) is symmetric, then \(q(x)=B(x,x)\) is a quadratic form. Conversely, over fields where \(2≠0\), a quadratic form determines its symmetric bilinear form by polarization. In real coordinates, write \(q(x)=x^TAx\) with \(A=A^T\).

Matrix recipe

• The coefficient of \(x_i^2\) is the diagonal entry \(a_{ii}\).
• The coefficient of \(x_i x_j\) for \(i≠ j\) is \(2a_{ij}\) when \(A\) is symmetric.
• So the off-diagonal entry is half the mixed-term coefficient.
• Only the symmetric part of a matrix contributes to \(x^TAx\), because \(x^TSx=0\) for skew-symmetric \(S\).

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Definiteness is about values on all nonzero vectors

Key idea

A quadratic form is positive definite when it is strictly positive away from zero. It is semidefinite when it never changes sign but can vanish on a nonzero vector. It is indefinite when it has both positive and negative directions.

Classification checklist

• Positive definite: \(q(x)>0\) for all \(x≠0\).
• Positive semidefinite: \(q(x)\ge0\) for all \(x\), with at least one nonzero zero direction if not definite.
• Negative definite: \(q(x)<0\) for all \(x≠0\).
• Negative semidefinite: \(q(x)\le0\) for all \(x\), with at least one nonzero zero direction if not definite.
• Indefinite: some vectors make \(q\) positive and some make \(q\) negative.

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Use minors or eigenvalues instead of guessing signs

Key idea

For a real symmetric matrix, eigenvalue signs classify the quadratic form. In dimension \(2\), the matrix \(A=\begin{pmatrix}a&b\\b&c\end{pmatrix}\) is positive definite exactly when \(a>0\) and \(ac-b^2>0\).

Sylvester criterion

• Positive definite symmetric matrix: all leading principal minors are positive.
• For \(2\times2\): \(a>0\) and \(\det A>0\).
• If \(\det A<0\) in dimension \(2\), the eigenvalues have opposite signs, so the form is indefinite.
• A zero determinant does not by itself prove semidefinite; check the remaining signs.

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Change variables to see rank and signature

Key idea

If \(A=A^T\), the spectral theorem gives \(A=QDQ^T\), and with \(y=Q^Tx\), \[x^TAx=y^TDy=\lambda_1y_1^2+\cdots+\lambda_ny_n^2.\] More generally, nonsingular changes of variables give congruent matrices \(P^TAP\). They may change eigenvalues, but they preserve the numbers of positive, negative, and zero squares.

Inertia facts

• The rank is the number of nonzero square coefficients in diagonal form.
• The signature \((n_+,n_-)\) counts positive and negative coefficients.
• The nullity counts zero coefficients.
• Sylvester law of inertia says these counts are invariant under nonsingular congruence.
• Similarity \(P^{-1}AP\) preserves eigenvalues; congruence \(P^TAP\) preserves inertia for symmetric forms.

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A real quadratic form remembers its symmetric bilinear form

Key idea

When \(q(x)=B(x,x)\) for a symmetric bilinear form over \(\mathbb{R}\), the polarization identity is \[B(x,y)=\frac{1}{2}\bigl(q(x+y)-q(x)-q(y)\bigr).\] Thus the diagonal data \(q(x)\) determines the whole symmetric bilinear form.

Polarization notes

• Positive definite symmetric \(B\) is an inner product.
• The matching norm is \(\|x\|=\sqrt{B(x,x)}=\sqrt{q(x)}\).
• A skew-symmetric real form contributes nothing to \(q(x)=B(x,x)\), because \(B(x,x)=0\).
• Polarization recovers the symmetric part, not a hidden skew-symmetric part.

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Most mistakes confuse matrices, signs, or change of basis

Common traps

• Mixed terms: the coefficient of \(xy\) is split across two symmetric off-diagonal entries.
• Bilinear is not symmetric: symmetry is extra, not part of the definition.
• Semidefinite is not definite: a nonzero zero direction prevents definiteness.
• Eigenvalue signs need symmetry: classify \(x^TAx\) using the symmetric part.
• Congruence is not similarity: congruence preserves inertia; similarity preserves eigenvalues.
• Skew-symmetric parts vanish on the diagonal: they do not affect \(q(x)=x^TAx\).

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Final recap

• Bilinear means linear in each argument separately.
• In coordinates, \(B(x,y)=x^TAy\).
• Symmetric bilinear forms correspond to symmetric matrices.
• A quadratic form is \(q(x)=B(x,x)\) or \(x^TAx\) with \(A=A^T\).
• Mixed coefficients are split across symmetric off-diagonal entries.
• Positive definite means \(q(x)>0\) for every nonzero \(x\).
• Semidefinite forms can vanish on nonzero vectors.
• Eigenvalue signs or Sylvester criterion classify real symmetric forms.
• Congruence preserves signature and rank; similarity preserves eigenvalues.
• Polarization recovers the symmetric bilinear form from \(q\).