Determinants

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.