Multilinear Algebra and Determinants

Prove that if $\beta$ is a bilinear form on $\mathbf{F}$, then there exists $c \in \mathbf{F}$ such that

for all $x, y \in \mathbf{F}$.

Let $n = \operatorname {dim} V$. Suppose $\beta$ is a bilinear form on $V$. Prove that there exist $\varphi_{1}, \ldots , \varphi_{n}, \tau_{1}, \ldots , \tau_{n} \in V^{\prime }$ such that

for all $u, v \in V$. This exercise shows that if $n = \operatorname {dim} V$, then every bilinear form on $V$ is of the form given by the last bullet point of Example 9.2.

Suppose $\beta : V \times V \rightarrow \mathbf{F}$ is a bilinear form on $V$ and also is a linear functional on $V \times V$. Prove that $\beta = 0$.

Suppose $V$ is a real inner product space and $\beta$ is a bilinear form on $V$. Show that there exists a unique operator $T \in \mathcal{L}(V)$ such that

for all $u, v \in V$. This exercise states that if $V$ is a real inner product space, then every bilinear form on $V$ is of the form given by the third bullet point in 9.2.

Suppose $\beta$ is a bilinear form on a real inner product space $V$ and $T$ is the unique operator on $V$ such that $\beta (u, v) = \left\langle u, T v \right\rangle$ for all $u, v \in V$ (see Exercise 4). Show that $\beta$ is an inner product on $V$ if and only if $T$ is an invertible positive operator on $V$.

Prove or give a counterexample: If $\rho$ is a symmetric bilinear form on $V$, then

is a subspace of $V$.

Explain why the proof of 9.13 (diagonalization of a symmetric bilinear form by an orthonormal basis on a real inner product space) fails if the hypothesis that $\mathbf{F} = \mathbb {R}$ is dropped.

Theorem 9.13: suppose $V$ is a real inner product space and $\rho$ is a symmetric bilinear form on $V$. Then $\rho$ has a diagonal matrix with respect to some orthonormal basis of $V$.

Proof of 9.13: let $f_{1}, \ldots , f_{n}$ be an orthonormal basis of $V$. Let $B = \mathcal{M}\left(\rho , \left(f_{1}, \ldots , f_{n}\right)\right)$. Then $B$ is a symmetric matrix (by 9.12). Let $T \in \mathcal{L}(V)$ be the operator such that $\mathcal{M}\left(T, \left(f_{1}, \ldots , f_{n}\right)\right) = B$. Thus $T$ is self-adjoint. The real spectral theorem (7.29) states that $T$ has a diagonal matrix with respect to some orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$. Let $C = \mathcal{M}\left(I, \left(e_{1}, \ldots , e_{n}\right), \left(f_{1}, \ldots , f_{n}\right)\right)$. Thus $C^{-1} B C$ is the matrix of $T$ with respect to the basis $e_{1}, \ldots , e_{n}$ (by 3.84). Hence $C^{-1} B C$ is a diagonal matrix. Now $\mathcal{M}\left(\rho , \left(e_{1}, \ldots , e_{n}\right)\right) = C^{\mathrm{t}} B C = C^{-1} B C$, where the first equality holds by 9.7 and the second equality holds because $C$ is a unitary matrix with real entries (which implies that $C^{-1} = C^{\mathrm{t}}$; see 7.57).

Find formulas for $\operatorname {dim} V_{\mathrm{sym}}^{(2)}$ and $\operatorname {dim} V_{\mathrm{alt}}^{(2)}$ in terms of $\operatorname {dim} V$.

Suppose that $n$ is a positive integer and $V = \left\{ p \in \mathcal{P}_{n}(\mathbb {R}): p(0) = p(1)\right\}$. Define $\alpha : V \times V \rightarrow \mathbb {R}$ by

Show that $\alpha$ is an alternating bilinear form on $V$.

Suppose that $n$ is a positive integer and

Define $\rho : V \times V \rightarrow \mathbb {R}$ by

Show that $\rho$ is a symmetric bilinear form on $V$.

Suppose $m$ is a positive integer. Show that $\operatorname {dim} V^{(m)} = (\operatorname {dim} V)^{m}$.

Suppose $n \geq 3$ and $\alpha : \mathbf{F}^{n} \times \mathbf{F}^{n} \times \mathbf{F}^{n} \rightarrow \mathbf{F}$ is defined by

Show that $\alpha$ is an alternating 3-linear form on $\mathbf{F}^{n}$.

Suppose $m$ is a positive integer and $\alpha$ is an $m$-linear form on $V$ such that $\alpha \left(v_{1}, \ldots , v_{m}\right) = 0$ whenever $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$ with $v_{j} = v_{j+1}$ for some $j \in \left\{ 1, \ldots , m-1\right\}$. Prove that $\alpha$ is an alternating $m$-linear form on $V$.

Prove or give a counterexample: If $\alpha \in V_{\text{alt }}^{(4)}$, then

is a subspace of $V^{4}$.

Suppose $m$ is a positive integer and $\beta$ is an $m$-linear form on $V$. Define an $m$-linear form $\alpha$ on $V$ by

for $v_{1}, \ldots , v_{m} \in V$. Explain why $\alpha \in V_{\text{alt }}^{(m)}$.

Suppose $m$ is a positive integer and $\beta$ is an $m$-linear form on $V$. Define an $m$-linear form $\alpha$ on $V$ by

for $v_{1}, \ldots , v_{m} \in V$. Explain why

for all $v_{1}, \ldots , v_{m} \in V$ and all $\left(k_{1}, \ldots , k_{m}\right) \in \operatorname {perm} m$.

Give an example of a nonzero alternating 2-linear form $\alpha$ on $\mathbb {R}^{3}$ and a linearly independent list $v_{1}, v_{2}$ in $\mathbb {R}^{3}$ such that $\alpha \left(v_{1}, v_{2}\right) = 0$.

Prove or give a counterexample: $S, T \in \mathcal{L}(V) \Longrightarrow \operatorname {det}\left(S+T\right) = \operatorname {det}\left(S\right)+\operatorname {det}\left(T\right)$.

Suppose the first column of a square matrix $A$ consists of all zeros except possibly the first entry $A_{1,1}$. Let $B$ be the matrix obtained from $A$ by deleting the first row and the first column of $A$. Show that $\operatorname {det}\left(A\right) = A_{1,1} \operatorname {det}\left(B\right)$.

Suppose $T \in \mathcal{L}(V)$ is nilpotent. Prove that $\operatorname {det}\left(I+T\right) = 1$.

Suppose $S \in \mathcal{L}(V)$. Prove that $S$ is unitary if and only if $\left|\operatorname {det}\left(S\right)\right| = \left\| S\right\| = 1$.

Suppose $A$ is a block upper-triangular matrix

where each $A_{k}$ along the diagonal is a square matrix. Prove that

Suppose $A = \left(\begin{array}{lll}v_{1} & \cdots & v_{n}\end{array}\right)$ is an $n$-by- $n$ matrix, with $v_{k}$ denoting the $k^{\text{th }}$ column of $A$. Show that if $\left(m_{1}, \ldots , m_{n}\right) \in$ perm $n$, then

Suppose $T \in \mathcal{L}(V)$ is invertible. Let $p$ denote the characteristic polynomial of $T$ and let $q$ denote the characteristic polynomial of $T^{-1}$. Prove that

for all nonzero $z \in \mathbf{F}$.

Suppose $T \in \mathcal{L}(V)$ is an operator with no eigenvalues (which implies that $\mathbf{F} = \mathbb {R}$). Prove that $\operatorname {det}\left(T\right) > 0$.

Suppose that $V$ is a real vector space of even dimension, $T \in \mathcal{L}(V)$, and $\operatorname {det}\left(T\right) < 0$. Prove that $T$ has at least two distinct eigenvalues.

Suppose $V$ is a real vector space of odd dimension and $T \in \mathcal{L}(V)$. Without using the minimal polynomial, prove that $T$ has an eigenvalue.

Prove or give a counterexample: If $\mathbf{F} = \mathbb {R}, T \in \mathcal{L}(V)$, and $\operatorname {det}\left(T\right) > 0$, then $T$ has a square root.

Suppose $S, T \in \mathcal{L}(V)$ and $S$ is invertible. Define $p: \mathbf{F} \rightarrow \mathbf{F}$ by

Prove that $p$ is a polynomial of degree $\operatorname {dim} V$ and that the coefficient of $z^{\operatorname {dim} V}$ in this polynomial is $\operatorname {det}\left(S\right)$.

Suppose $\mathbf{F} = \mathbb {C}, T \in \mathcal{L}(V)$, and $n = \operatorname {dim} V > 2$. Let $\lambda_{1}, \ldots , \lambda_{n}$ denote the eigenvalues of $T$, with each eigenvalue included as many times as its multiplicity.

Suppose $V$ is an inner product space and $T$ is a positive operator on $V$. Prove that

Suppose $V$ is an inner product space and $T \in \mathcal{L}(V)$. Use the polar decomposition to give a proof that

that is different from the proof given earlier (see 9.60).

Suppose $T \in \mathcal{L}(V)$. Define $g: \mathbf{F} \rightarrow \mathbf{F}$ by $g(x) = \operatorname {det}\left(I+x T\right)$. Show that $g^{\prime }(0) = \operatorname {tr} T$.

Suppose $a, b, c$ are positive numbers. Find the volume of the ellipsoid

by finding a set $\Omega \subseteq \mathbb {R}^{3}$ whose volume you know and an operator $T$ on $\mathbb {R}^{3}$ such that $T(\Omega )$ equals the ellipsoid above.

Suppose that $A$ is an invertible square matrix. Prove that Hadamard's inequality (9.66) is an equality if and only if each column of $A$ is orthogonal to the other columns.

Theorem 9.66 (Hadamard's inequality): suppose $A$ is an $n$-by-$n$ matrix. Let $v_{1}, \ldots , v_{n}$ denote the columns of $A$. Then $\left|\operatorname {det}\left(A\right)\right| \leq \prod_{k = 1}^{n}\left\| v_{k}\right\|$.

Suppose $V$ is an inner product space, $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$, and $T \in \mathcal{L}(V)$ is a positive operator.

Suppose $A$ is an $n$-by- $n$ matrix, and suppose $c$ is such that $\left|A_{j, k}\right| \leq c$ for all $j, k \in \left\{ 1, \ldots , n\right\}$. Prove that

The formula for the determinant of a matrix (9.46) shows that $\left|\operatorname {det}\left(A\right)\right| \leq c^{n} n$ !. However, the estimate given by this exercise is much better. For example, if $c = 1$ and $n = 100$, then $c^{n} n!\approx 10^{158}$, but the estimate given by this exercise is the much smaller number $10^{100}$. If $n$ is an integer power of 2, then the inequality above is sharp and cannot be improved.

Suppose $n$ is a positive integer and $\delta : \mathbb {C}^{n, n} \rightarrow \mathbb {C}$ is a function such that

for all $A, B \in \mathbb {C}^{n, n}$ and $\delta (A)$ equals the product of the diagonal entries of $A$ for each diagonal matrix $A \in \mathbb {C}^{n, n}$. Prove that

for all $A \in \mathbb {C}^{n, n}$. Recall that $\mathbb {C}^{n, n}$ denotes set of $n$-by-n matrices with entries in $\mathbb {C}$. This exercise shows that the determinant is the unique function defined on square matrices that is multiplicative and has the desired behavior on diagonal matrices. This result is analogous to Exercise 10 in Section 8D, which shows that the trace is uniquely determined by its algebraic properties.

Suppose $v \in V$ and $w \in W$. Prove that $v \otimes w = 0$ if and only if $v = 0$ or $w = 0$.

Give an example of six distinct vectors $v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}$ in $\mathbb {R}^{3}$ such that

but none of $v_{1} \otimes w_{1}, v_{2} \otimes w_{2}, v_{3} \otimes w_{3}$ is a scalar multiple of another element of this list.

Suppose that $v_{1}, \ldots , v_{m}$ is a linearly independent list in $V$. Suppose also that $w_{1}, \ldots , w_{m}$ is a list in $W$ such that

Prove that $w_{1} = \cdots = w_{m} = 0$.

Suppose $\operatorname {dim} V > 1$ and $\operatorname {dim} W > 1$. Prove that

is not a subspace of $V \otimes W$. This exercise implies that if $\operatorname {dim} V > 1$ and $\operatorname {dim} W > 1$, then

Suppose $m$ and $n$ are positive integers. For $v \in \mathbf{F}^{m}$ and $w \in \mathbf{F}^{n}$, identify $v \otimes w$ with an $m$-by- $n$ matrix as in Example 9.76. With that identification, show that the set

is the set of $m$-by- $n$ matrices (with entries in $\mathbf{F}$) that have rank at most one.

Suppose $m$ and $n$ are positive integers. Give a description, analogous to Exercise 5, of the set of $m$-by- $n$ matrices (with entries in $\mathbf{F}$) that have rank at most two.

Suppose $\operatorname {dim} V > 2$ and $\operatorname {dim} W > 2$. Prove that

Suppose $v_{1}, \ldots , v_{m} \in V$ and $w_{1}, \ldots , w_{m} \in W$ are such that

Suppose that $U$ is a vector space and $\Gamma : V \times W \rightarrow U$ is a bilinear map. Show that

Suppose $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$. Prove that there exists a unique operator on $V \otimes W$ that takes $v \otimes w$ to $S v \otimes T w$ for all $v \in V$ and $w \in W$.

Suppose $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$. Prove that $S \otimes T$ is an invertible operator on $V \otimes W$ if and only if both $S$ and $T$ are invertible operators. Also, prove that if both $S$ and $T$ are invertible operators, then $(S \otimes T)^{-1} = S^{-1} \otimes T^{-1}$, where we are using the notation from the comment after Exercise 9.

Suppose $V$ and $W$ are inner product spaces. Prove that if $S \in \mathcal{L}(V)$ and $T \in \mathcal{L}(W)$, then $(S \otimes T)^{*} = S^{*} \otimes T^{*}$, where we are using the notation from the comment after Exercise 9.

Suppose that $V_{1}, \ldots , V_{m}$ are finite-dimensional inner product spaces. Prove that there is a unique inner product on $V_{1} \otimes \cdots \otimes V_{m}$ such that

for all $\left(v_{1}, \ldots , v_{m}\right)$ and $\left(u_{1}, \ldots , u_{m}\right)$ in $V_{1} \times \cdots \times V_{m}$. Note that the equation above implies that

for all $\left(v_{1}, \ldots , v_{m}\right) \in V_{1} \times \cdots \times V_{m}$.

Suppose that $V_{1}, \ldots , V_{m}$ are finite-dimensional inner product spaces and $V_{1} \otimes \cdots \otimes V_{m}$ is made into an inner product space using the inner product from Exercise 12. Suppose $e_{1}^{k}, \ldots , e_{n_{k}}^{k}$ is an orthonormal basis of $V_{k}$ for each $k = 1, \ldots , m$. Show that the list

is an orthonormal basis of $V_{1} \otimes \cdots \otimes V_{m}$.