(a) Find the Gram matrix corresponding to each of the...

(a) Find the Gram matrix corresponding to each of the following sets of vectors using the Euclidean dot product on $\mathbb{R}^n$. (i) $\left(\begin{array}{r}-1 \\ 3\end{array}\right),\left(\begin{array}{l}0 \\ 2\end{array}\right)$, (ii) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{r}-2 \\ 3\end{array}\right),\left(\begin{array}{l}-1 \\ -1\end{array}\right)$, (iii) $\left(\begin{array}{r}2 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}-3 \\ 0 \\ 2\end{array}\right)$, (iv) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$, (v) $\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ -1 \\ 1\end{array}\right)$, (vi) $\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0 \\ 1\end{array}\right)$, (vii) $\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ -4 \\ 3\end{array}\right),\left(\begin{array}{r}-1 \\ 3 \\ -1 \\ -2\end{array}\right)$, (viii) $\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -3 \\ 0\end{array}\right)$. (b) Which are positive definite? (c) If the matrix is positive semi-definite, find all its null directions.

(a) Find the Gram matrix corresponding to each of the following sets of vectors using the Euclidean dot product on $\mathbb{R}^n$.
(i) $\left(\begin{array}{r}-1 \\ 3\end{array}\right),\left(\begin{array}{l}0 \\ 2\end{array}\right)$,
(ii) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{r}-2 \\ 3\end{array}\right),\left(\begin{array}{l}-1 \\ -1\end{array}\right)$,
(iii) $\left(\begin{array}{r}2 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}-3 \\ 0 \\ 2\end{array}\right)$,
(iv) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$,
(v) $\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ -1 \\ 1\end{array}\right)$,
(vi) $\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0 \\ 1\end{array}\right)$,
(vii)
$\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ -4 \\ 3\end{array}\right),\left(\begin{array}{r}-1 \\ 3 \\ -1 \\ -2\end{array}\right)$,
(viii)
$\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -3 \\ 0\end{array}\right)$.
(b) Which are positive definite? (c) If the matrix is positive semi-definite, find all its null directions.

Key Concepts

Null Directions (Null Space)

The null directions of a Gram matrix are the nonzero vectors x for which the quadratic form yields zero (i.e., x^T G x = 0) or, equivalently, for which Gx = 0. These null directions correspond to the directions in the vector space that are 'invisible' to the inner product defined by the set of vectors. In the context of Gram matrices, they indicate the presence of linear dependencies among the vectors, and finding all null directions is essential for understanding the rank and stability of the associated matrix.

Positive Semi-Definiteness

A matrix is positive semi-definite if for all vectors x, the quadratic form x^T G x is non-negative. This property is inherent in Gram matrices, but if the underlying vectors are linearly dependent, the matrix will only be semi-definite. Understanding whether a Gram matrix is only semi-definite helps in identifying cases where additional analysis is needed, such as determining null directions or degeneracies in the data.

Positive Definiteness

A matrix is positive definite if for all nonzero vectors x, the quadratic form x^T G x is strictly positive. This concept is crucial when analyzing Gram matrices since positive definiteness is equivalent to the original set of vectors being linearly independent. It ensures that the inner product structure induces a proper norm, making it an important property in numerical methods and optimization.

Euclidean Dot Product

The Euclidean dot product is a specific inner product on ?^n defined by summing the product of corresponding components of two vectors. It is used to measure lengths and angles, and it serves as the basis for constructing Gram matrices. Its properties, such as commutativity, distributivity, and the fact that the dot product of a vector with itself gives the square of its norm, make it fundamental in many geometric and algebraic formulations.

Gram Matrix

The Gram matrix is constructed by taking a set of vectors and computing all pairwise inner products (using the dot product in Euclidean space). Each entry of the matrix is given by g_ij = ?v_i, v_j?. This matrix encapsulates the geometric relationships between the vectors, such as angles and lengths, and is central to many applications in linear algebra and data analysis.

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