Nikiya Anton Bettey 2021: 09-26-2021-2128

Monday, September 27, 2021

09-26-2021-2128 - trace

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In linear algebra, the trace of a square matrix $A$ , denoted $tr(A)$ ,^[1]^[2] is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of $A$ .

The trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities), and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. The trace is only defined for a square matrix ( $n \times n$ ).

The trace is related to the derivative of the determinant (see Jacobi's formula).

Trace of a Kronecker product[edit]

The trace of the Kronecker product of two matrices is the product of their traces:

\operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).

Full characterization of the trace[edit]

The following three properties:

{\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}

characterize the trace completely in the sense that follows. Let $f$ be a linear functional on the space of square matrices satisfying $f (xy) = f (yx)$ . Then $f$ and $tr$ are proportional.^{[note 2]}

Similarity invariance[edit]

The trace is similarity-invariant, which means that for any square matrix $A$ and any invertible matrix $P$ of the same dimensions, the matrices $A$ and $P -1 AP$ have the same trace. This is because

\operatorname {tr} \left(\mathbf {P} ^{-1}\mathbf {A} \mathbf {P} \right)=\operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} \left(\mathbf {A} \left(\mathbf {P} \mathbf {P} ^{-1}\right)\right)=\operatorname {tr} (\mathbf {A} ).

Trace of product of symmetric and skew-symmetric matrix[edit]

If $A$ is symmetric and $B$ is skew-symmetric, then

\operatorname {tr} (\mathbf {A} \mathbf {B} )=0

Relation to eigenvalues[edit]

Trace of the identity matrix[edit]

The trace of the $n \times n$ identity matrix is the dimension of the space, namely $n$ .^[1]

\operatorname {tr} \left(\mathbf {I} _{n}\right)=n

This leads to generalizations of dimension using trace.

Trace of an idempotent matrix[edit]

The trace of an idempotent matrix $A$ (a matrix for which $A 2 = A$ ) is equal to the rank of $A$ .

Trace of a nilpotent matrix[edit]

The trace of a nilpotent matrix is zero.

When the characteristic of the base field is zero, the converse also holds: if $tr(A k) = 0$ for all $k$ , then $A$ is nilpotent.

When the characteristic $n > 0$ is positive, the identity in $n$ dimensions is a counterexample, as $\operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0$ , but the identity is not nilpotent.

Trace equals sum of eigenvalues[edit]

More generally, if

f(x)=\prod _{i=1}^{k}\left(x-\lambda _{i}\right)^{d_{i}}

is the characteristic polynomial of a matrix $A$ , then

$\operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{k}d_{i}\lambda _{i}$

that is, the trace of a square matrix equals the sum of the eigenvalues counted with multiplicities.

https://en.wikipedia.org/wiki/Trace_(linear_algebra)

Nikiya Anton Bettey 2021

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