Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). How many zeros can I get?

Equivalent formulation: what is the largest number $k=k(n)$ such that every real $n\times n$ matrix is orthogonally equivalent to a matrix with at least $k$ zero entries?

Updated with partial results

  1. Since the orthogonal group $O(n)$ has dimension $\frac12 n(n-1)$, it follows that $$k(n)\le \frac12 n(n-1) \tag{1}$$ Indeed, the set of all matrices with $k$ particular entries set to $0$ is an $(n^2-k)$-dimensional vector space, hence its orbit under $A\mapsto O^TAO$ ($O$ orthogonal) has at most $n^2-k+\frac12 n(n-1)$ dimensions. (Hausdorff dimension can be used here to make this rigorous.)

  2. On the other hand, the answer by Omnomnomnom gives a lower bound $$k(n)\ge \left\lceil \frac12 n(n-2) \right\rceil \tag{2}$$ A gap remains here. It is easy to see that $k(2)=0$ (e.g., a generic rotation of the plane has full matrix in every orthonormal basis), so (2) is sharp in this case. For $n=3$, the inequalities give $2\le k(3)\le 3$; which one is sharp?

  3. The same question could be asked for complex matrices and unitary equivalence. Let $k_{\mathbb C}(n)$ denote the analog of $k(n)$ for this case. Since the real dimension of $U(n)$ is $n^2$, it follows that $$k_{\mathbb C}(n) \le \left\lfloor \frac12 n^2 \right\rfloor\tag{3}$$ In the opposite direction, Omnomnomnom pointed out the lower bound $$k_{\mathbb C}(n)\ge \frac12 n(n-1) \tag{4}$$ There is a gap here too.

  • $\begingroup$ I think you are going to get $k(n)=\frac12n(n-1)$ for $n$ odd and $k(n)=\frac1n(n-1)-1$ for $n$ even. As mentioned in answers already, you can transform to block upper triangular form, where the blocks on the diagonal are 1x1 or 2x2. You can't remove all nonzero entries below the diagonal (unless all e-vals are real), but you can introduce some additional zeros above the diagonal, as in Post No Bill's answer to get to the max $k(n)$ I just mentioned. For even $k$, if all eigenvalues are complex, it gets difficult to add the final 0 above the diagonal, so I think you might lose 1 from $k$. $\endgroup$ Dec 10 '13 at 12:08
  • $\begingroup$ Actually, I can see how you can add the final zero to get the optimal $k(n)=\frac12n(n-1)$ for all $n > 2$ (using the singular value decomposition on one of the 2x2 upper triangular blocks). $\endgroup$ Dec 10 '13 at 12:59
  • $\begingroup$ And for complex case, you can always make it upper triangular, which is best possible, giving $k_c(n)=\frac12n(n-1)$. $\endgroup$ Dec 10 '13 at 13:26
  • $\begingroup$ @GeorgeLowther I do not see a proof that (i) sufficiently many zeros can be introduced above diagonal in the real case; (ii) there can be no improvement on the upper triangular form in the complex cases. $\endgroup$ Dec 10 '13 at 13:31
  • $\begingroup$ For the real case, I can post an answer when I get time, maybe tonight. For the complex case, $U(n)$ quotiented out by the diagonal unitary matrices (which don't change the number of nonzero entries in your matrix) has dimension $n(n-1)$, which is the same as the dimension of the strict lower triangular part of your matrix, so you can't eliminate any further entries. $\endgroup$ Dec 10 '13 at 14:37

We have $k_{\mathbb{C}}(n)=k(n)=\frac12n(n-1)$ for all $n > 2$ (for $n=2$, we have $k_{\mathbb{C}}(2)=1$, $k(2)=0$).

First, I'll show that these are upper bounds before showing that they can be attained. I'll use the fact that a smooth map from an $m$ dimensional manifold (or finite union of $m$ dimensional manifolds) to an $n$ dimensional manifold cannot be onto unless $n\le m$. It was noted in the question that $k(n)\le\frac12n(n-1)$, so I'll just look at the complex case. The space ${\rm U}(n)$ of $n\times n$ unitary matrices has dimension $n^2$, and the space ${\rm GL}_{\mathbb{C}}(n)$ of complex $n\times n$ matrices has dimension $2n^2$. If $U$ is unitary such that $U^HAU$ has $k$ zero entries, then $V^HAV$ also has $k$ zero entries, where $V=DU$ for a diagonal unitary matrix $D$. For example, it is always possible to choose $D$ such that the first nonzero element of each row of $V$ is real. Letting $R$ be the space of $n\times n$ unitary matrices whose first nonzero element in each row is real, this has dimension $n(n-1)$. Letting $S$ be the $n\times n$ complex matrices with at least $k$ zero entries, this has dimension $2(n^2-k)$. We require the map $R\times S\to {\rm GL}_{\mathbb{C}}(n)$, $(U,M)\mapsto UMU^H$ to be onto. So, $$ {\rm dim}(R\times S)=n(n-1)+2(n^2-k)\ge{\rm dim}(GL_{\mathbb{C}}(n))=2n^2, $$ so, again, $k\le\frac12n(n-1)$.

Now, we can prove the lower bound. It was noted in the question that $k_{\mathbb{C}}\ge\frac12n(n-1)$, as all complex matrices can be put in upper triangular form by a Schur decomposition. So, only the real case remains. For positive integers $n_1+n_2+\cdots+n_m=n$, we can express an $n\times n$ matrix $A$ in block form $$ A=\pmatrix{ A_{11} & A_{12} &\cdots&&A_{1m}\\ A_{21}&A_{22}&\cdots&&A_{2m}\\ \vdots&&\ddots&&\vdots\\ \\ A_{m1}&A_{m2}&\cdots&& A_{mm} } $$ where $A_{ij}$ is an $n_i\times n_j$ real matrix. As noted by Omnomnomnom, using the real Schur form, then replacing $A$ by $Q^TAQ$ for orthogonal $Q$, this can be done so that it is block upper triangular ($A_{ij}=0$ for $i > j$) and such that each $n_i$ is 1 or 2. We can take $n_1=n_2=\cdots=n_r=2$ and $n_{r+1}=n_{r+2}=\cdots=n_m=1$ (where $r\in\{0,1,\ldots,m\}$ is the number of complex-conjugate pairs of eigenvalues of $A$). The nonzero below diagonal terms of $A$ then correspond to the below diagonal terms of the block diagonal elements $A_{ii}$. There are $r$ of these, so $A$ has at least $\frac12n(n-1)-r$ zero entries below the diagonal. We cannot remove these remaining $r$ below-diagonal nonzero entries, but we can introduce an additional $r$ zeros above the diagonal. Note first that if $Q$ is a real matrix in block form $Q=(Q_{ij})_{i,j=1,\ldots,n}$ with $Q_{ij}=0$ for $i\not=j$ (i.e., block-diagonal) with each $Q_{ii}$ orthogonal, then $Q$ is orthogonal. Furthermore, $B=Q^TAQ$ has block form $B=(B_{ij})$ with $B_{ij}=Q_{ii}^TA_{ij}Q_{jj}$. So, $B$ is also block upper triangular.

(1) For any $k < m$ with $n_k=2$, there is an orthogonal block-diagonal matrix $Q$ such that $B=Q^TAQ$ has block entries $B_{ij}=A_{ij}$ for $i,j\not=k$ and $B_{km}$ has a zero entry.

To show this, let $v\in\mathbb{R}^2$ be the first column of $A_{km}$ and $R$ be a $2\times2$ rotation matrix with $R^Tv=(\lVert v\rVert,0)^T$. Then, $R^TA_{km}$ is upper triangular and letting $Q$ be the block-diagonal matrix with $Q_{kk}=R$ and all other block diagonal elements being the identity gives the result.

As long as $r < m$ or, equivalently, $A$ has at least one real eigenvalue, we can apply this so that $A_{rm}$ has a zero entry, then so that $A_{r-1,m}$ has a zero entry, and so on to end up with each $A_{im}$ having a zero entry for $i\le r$. If $r=m$ or, equivalently, $A$ has no real eigenvalues, we can use the following.

(2) If $i < j$ is such that $n_i=n_j=2$ then there exists an orthogonal block-diagonal matrix $Q$ such that $B=Q^TAQ$ has block entry $B_{ij}$ being diagonal.

By the singular value decomposition, there exists $2\times2$ orthogonal matrices $R,S$ with $R^TA_{ij}S$ being diagonal. Then, letting $Q$ be the block-diagonal matrix with $Q_{ii}=R$, $Q_{jj}=S$ and all other diagonal entries being the identity gives the result.

Now in the case where $n > 2$ and $A$ has all complex eigenvalues (so $r=m\ge2$), we can apply (2) to make $A_{r-1,r}$ diagonal so that it has two zero entries. Iteratively applying (1) with $i=r-2,r-3,\ldots$ introduces a zero entry in each of the blocks $A_{im}$ for $i\le r-2$. This puts $A$ into upper block triangular form with $r$ zeros above the diagonal.

  • $\begingroup$ This is a neat argument. Thanks for writing it out in detail. $\endgroup$ Dec 11 '13 at 1:49

With Schur triangularization, you can get the slightly improved result of $\frac 12 n(n-1)$ zero-entries in the complex case, assuming the matrix has exclusively real eigenvalues.

For a real matrix with complex eigenvalues, we can always put the matrix into its "real Schur form", which is a "block-upper-triangular" form. That is, for $A \in \mathbb{R}^{n\times n}$, there exists an orthogonal matrix $Q$ for which $$ Q^TAQ = \pmatrix{ A_1 &&&*\\ &A_2&&\\ &&\ddots&\\ 0&&& A_k } $$ Where each $A_i$ is $1\times 1$ or a $2\times 2$ real matrix with a conjugate pair of complex eigenvalues.

This means that for any $A$, we can always choose an orthogonally equivalent matrix with at least $\frac 12 n(n-1) - n/2 = \frac 12 n(n-2)$ zero entries, which is to say at least $\lceil \frac 12 n(n-2) \rceil$ zero entries.

I hope you find this useful.

It may be possible to generalize your latest result.

For any odd $n$, any $T \in \mathbb{R}^{n\times n}$ must have some eigenvector $v_1$.

So, consider any transformation $A$ with odd dimension. We can write $T = Q^TAT$ of the above form, insisting that $A_1$ be a $1 \times 1$ matrix. That is, for some $a \in \mathbb{R}$, we can write $$ Q^TAQ = \pmatrix{ a &&&*\\ 0&A_2&&\\ \vdots&&\ddots&\\ 0&0&& A_k } $$ Note that the first column of $Q$ (call it $q_1$) is an eigenvector, and that $a$ is the corresponding eigenvalue. If $T$ is non-invertible (which is true if and only if $A$ is invertible), we can insist $a=0$, and we have our extra zero-entry as desired.

If $T$ is non-invertible, I think we might be able to set $v_1 = q_1$, and choose a new basis $v_2,\dots,v_n$ covering $P =$ the span of $q_2,\dots,q_n$. Maybe there's some way to choose $v_2,\dots,v_n$ so that $v_2 \in P \cap T(P)$, and we maintain this block upper-triangular form.

Food for thought, I suppose.

  • $\begingroup$ In writing my answer, I had forgotten you were looking at real matrices. I have updated my answer accordingly $\endgroup$ Dec 1 '13 at 19:41
  • $\begingroup$ Thanks, this is indeed useful. $\endgroup$ Dec 4 '13 at 5:11

Here is a proof that $k(3)=3$.

Let $T:\mathbb R^3\to\mathbb R^3$ be a linear operator. If the kernel of $T$ is nontrivial, choose a vector in the kernel as one of basis element; this creates $3$ zeros in the matrix already. Suppose $T$ is invertible. Since the space is odd-dimensional, $T$ has at least one eigenvector (the characteristic polynomial has at least one real root). Let $v_1$ be a normalized eigenvector of $T$. Let $P$ be the 2-dimensional subspace orthogonal to $v_1$. Since $T^{-1}P$ is also 2-dimensional, the intersection $P\cap T^{-1}P$ contains a line. Let $v_2$ be a unit vector on the line, and notice that $Tv_2 \perp v_1$. Finally, complete the basis with $v_3=v_1\times v_2$.

In the basis $v_1,v_2,v_3$ the matrix of $T$ has the form $$\begin{pmatrix} * & 0 & * \\ 0 & * & * \\ 0 & * & * \end{pmatrix} $$ Thus, $k(3)\ge 3$. The reverse inequality follows from (1) in the question.

  • 1
    $\begingroup$ This looks right to me, I must have overlooked something in my (now deleted) answer. $\endgroup$ Dec 7 '13 at 20:50
  • $\begingroup$ I don't understand why $v_1^T T v_2 = 0$. In general, $v_2$ could map almost anywhere in $TP$, and most such directions are not in $P$. Having said that, there certainly is a vector in $P$ that maps into $P$, so I think your result is correct. $\endgroup$
    – apt1002
    Dec 8 '13 at 4:23
  • $\begingroup$ @apt1002 Thanks for noticing the bug; fixed. $\endgroup$ Dec 8 '13 at 4:29

If all the eigenvalues are distinct, then the matrix is fully diagonalizable so that there would be $n^2 - n = n(n-1)$ zeros. If some eigenvalues are identical, then the matrix is only block-diagonalizable; if there are $m\le n$ distinct eigenvalues with $n_1$ to $n_m$ being the number of vectors with that eigenvalue, then the blocks would be $n_i \times n_i$ squares and we could guarantee at least $n^2 - \sum_{i=1}^mn_i^2$ zeros.

Unfortunately, the worst case for this line of reasoning is when all eigenvalues are identical. In that case it cannot prove any zeros at all.

The $k$-numbers under investigation here can be used to tighten the above result. Since the restriction of the linear operator to the $i$th block must still be able to have $k(n_i)$ zeros forced forced into it, we can guarantee at least $n^2 - \sum_{i=1}^mn_i^2 + \sum_{i=1}^mk(n_i)$ zeros, which will be a higher number if at least one $n_i\ge 3$ (so that $k(n_i)>0$).

If we think of the real matrix as a complex matrix, then its Jordan Normal Form will have at most $n$ non-zero entries on the diagonal and $\sum_{i=1}^m(n_i-1) = n-m$ non-zero entries on the superdiagonal. Therefore, we would get no more than $2n-m$ non-zero entries and at least $n^2 - 2n + m$ zeros. However, the JNF of a real matrix is not necessarily real itself, unless all the eigenvalues are also real, so getting this many zeros in the general case would require allowing complex values (on the main diagonal only) in the JNF.

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    $\begingroup$ Note that this question is specifically about unitary/orthogonal similarity, which prohibits the general use of Jordan canonical form. Even if $A$ has distinct eigenvalues, it is only unitarily diagonalizable if it is normal. $\endgroup$ Dec 10 '13 at 11:52

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