diff --git a/mat/band.go b/mat/band.go
index 20225109..72ebefdd 100644
--- a/mat/band.go
+++ b/mat/band.go
@@ -188,6 +188,13 @@ func (b *BandDense) RawBand() blas64.Band {
 	return b.mat
 }
 
+// SetRawBand sets the underlying blas64.Band used by the receiver.
+// Changes to elements in the receiver following the call will be reflected
+// in the input.
+func (b *BandDense) SetRawBand(mat blas64.Band) {
+	b.mat = mat
+}
+
 // DiagView returns the diagonal as a matrix backed by the original data.
 func (b *BandDense) DiagView() Diagonal {
 	n := min(b.mat.Rows, b.mat.Cols)
diff --git a/mat/dense.go b/mat/dense.go
index be4ba8bb..87b1105c 100644
--- a/mat/dense.go
+++ b/mat/dense.go
@@ -134,16 +134,6 @@ func (m *Dense) Zero() {
 	}
 }
 
-// untranspose untransposes a matrix if applicable. If a is an Untransposer, then
-// untranspose returns the underlying matrix and true. If it is not, then it returns
-// the input matrix and false.
-func untranspose(a Matrix) (Matrix, bool) {
-	if ut, ok := a.(Untransposer); ok {
-		return ut.Untranspose(), true
-	}
-	return a, false
-}
-
 // isolatedWorkspace returns a new dense matrix w with the size of a and
 // returns a callback to defer which performs cleanup at the return of the call.
 // This should be used when a method receiver is the same pointer as an input argument.
@@ -552,3 +542,17 @@ func (m *Dense) Augment(a, b Matrix) {
 	w := m.Slice(0, br, ac, ac+bc).(*Dense)
 	w.Copy(b)
 }
+
+// Trace returns the trace of the matrix. The matrix must be square or Trace
+// will panic.
+func (m *Dense) Trace() float64 {
+	if m.mat.Rows != m.mat.Cols {
+		panic(ErrSquare)
+	}
+	// TODO(btracey): could use internal asm sum routine.
+	var v float64
+	for i := 0; i < m.mat.Rows; i++ {
+		v += m.mat.Data[i*m.mat.Stride+i]
+	}
+	return v
+}
diff --git a/mat/matrix.go b/mat/matrix.go
index df217d9d..444d0445 100644
--- a/mat/matrix.go
+++ b/mat/matrix.go
@@ -207,6 +207,73 @@ type ColNonZeroDoer interface {
 	DoColNonZero(j int, fn func(i, j int, v float64))
 }
 
+// untranspose untransposes a matrix if applicable. If a is an Untransposer, then
+// untranspose returns the underlying matrix and true. If it is not, then it returns
+// the input matrix and false.
+func untranspose(a Matrix) (Matrix, bool) {
+	if ut, ok := a.(Untransposer); ok {
+		return ut.Untranspose(), true
+	}
+	return a, false
+}
+
+// untransposeExtract returns an untransposed matrix in a built-in matrix type.
+//
+// The untransposed matrix is returned unaltered if it is a built-in matrix type.
+// Otherwise, if it implements a Raw method, an appropriate built-in type value
+// is returned holding the raw matrix value of the input. If neither of these
+// is possible, the untransposed matrix is returned.
+func untransposeExtract(a Matrix) (Matrix, bool) {
+	ut, trans := untranspose(a)
+	switch m := ut.(type) {
+	case *DiagDense, *SymBandDense, *TriBandDense, *BandDense, *TriDense, *SymDense, *Dense:
+		return m, trans
+	// TODO(btracey): Add here if we ever have an equivalent of RawDiagDense.
+	case RawSymBander:
+		rsb := m.RawSymBand()
+		if rsb.Uplo != blas.Upper {
+			return ut, trans
+		}
+		var sb SymBandDense
+		sb.SetRawSymBand(rsb)
+		return &sb, trans
+	case RawTriBander:
+		rtb := m.RawTriBand()
+		if rtb.Diag == blas.Unit {
+			return ut, trans
+		}
+		var tb TriBandDense
+		tb.SetRawTriBand(rtb)
+		return &tb, trans
+	case RawBander:
+		var b BandDense
+		b.SetRawBand(m.RawBand())
+		return &b, trans
+	case RawTriangular:
+		rt := m.RawTriangular()
+		if rt.Diag == blas.Unit {
+			return ut, trans
+		}
+		var t TriDense
+		t.SetRawTriangular(rt)
+		return &t, trans
+	case RawSymmetricer:
+		rs := m.RawSymmetric()
+		if rs.Uplo != blas.Upper {
+			return ut, trans
+		}
+		var s SymDense
+		s.SetRawSymmetric(rs)
+		return &s, trans
+	case RawMatrixer:
+		var d Dense
+		d.SetRawMatrix(m.RawMatrix())
+		return &d, trans
+	default:
+		return ut, trans
+	}
+}
+
 // TODO(btracey): Consider adding CopyCol/CopyRow if the behavior seems useful.
 // TODO(btracey): Add in fast paths to Row/Col for the other concrete types
 // (TriDense, etc.) as well as relevant interfaces (RowColer, RawRowViewer, etc.)
@@ -803,44 +870,28 @@ func Sum(a Matrix) float64 {
 	return sum
 }
 
+// A Tracer can compute the trace of the matrix. Trace must panic if the
+// matrix is not square.
+type Tracer interface {
+	Trace() float64
+}
+
 // Trace returns the trace of the matrix. Trace will panic if the
 // matrix is not square.
 func Trace(a Matrix) float64 {
+	m, _ := untransposeExtract(a)
+	if t, ok := m.(Tracer); ok {
+		return t.Trace()
+	}
 	r, c := a.Dims()
 	if r != c {
 		panic(ErrSquare)
 	}
-
-	aU, _ := untranspose(a)
-	switch m := aU.(type) {
-	case RawMatrixer:
-		rm := m.RawMatrix()
-		var t float64
-		for i := 0; i < r; i++ {
-			t += rm.Data[i*rm.Stride+i]
-		}
-		return t
-	case RawTriangular:
-		rm := m.RawTriangular()
-		var t float64
-		for i := 0; i < r; i++ {
-			t += rm.Data[i*rm.Stride+i]
-		}
-		return t
-	case RawSymmetricer:
-		rm := m.RawSymmetric()
-		var t float64
-		for i := 0; i < r; i++ {
-			t += rm.Data[i*rm.Stride+i]
-		}
-		return t
-	default:
-		var t float64
-		for i := 0; i < r; i++ {
-			t += a.At(i, i)
-		}
-		return t
+	var v float64
+	for i := 0; i < r; i++ {
+		v += a.At(i, i)
 	}
+	return v
 }
 
 func min(a, b int) int {
diff --git a/mat/symband.go b/mat/symband.go
index a6b71030..add9a807 100644
--- a/mat/symband.go
+++ b/mat/symband.go
@@ -145,6 +145,18 @@ func (s *SymBandDense) RawSymBand() blas64.SymmetricBand {
 	return s.mat
 }
 
+// SetRawSymBand sets the underlying blas64.SymmetricBand used by the receiver.
+// Changes to elements in the receiver following the call will be reflected
+// in the input.
+//
+// The supplied SymmetricBand must use blas.Upper storage format.
+func (s *SymBandDense) SetRawSymBand(mat blas64.SymmetricBand) {
+	if mat.Uplo != blas.Upper {
+		panic("mat: blas64.SymmetricBand does not have blas.Upper storage")
+	}
+	s.mat = mat
+}
+
 // Zero sets all of the matrix elements to zero.
 func (s *SymBandDense) Zero() {
 	for i := 0; i < s.mat.N; i++ {
diff --git a/mat/symmetric.go b/mat/symmetric.go
index c859b534..2ea5bdb0 100644
--- a/mat/symmetric.go
+++ b/mat/symmetric.go
@@ -113,13 +113,14 @@ func (s *SymDense) RawSymmetric() blas64.Symmetric {
 
 // SetRawSymmetric sets the underlying blas64.Symmetric used by the receiver.
 // Changes to elements in the receiver following the call will be reflected
-// in b. SetRawSymmetric will panic if b is not an upper-encoded symmetric
-// matrix.
-func (s *SymDense) SetRawSymmetric(b blas64.Symmetric) {
-	if b.Uplo != blas.Upper {
+// in the input.
+//
+// The supplied Symmetric must use blas.Upper storage format.
+func (s *SymDense) SetRawSymmetric(mat blas64.Symmetric) {
+	if mat.Uplo != blas.Upper {
 		panic(badSymTriangle)
 	}
-	s.mat = b
+	s.mat = mat
 }
 
 // Reset zeros the dimensions of the matrix so that it can be reused as the
@@ -514,6 +515,16 @@ func (s *SymDense) SliceSym(i, k int) Symmetric {
 	return &v
 }
 
+// Trace returns the trace of the matrix.
+func (s *SymDense) Trace() float64 {
+	// TODO(btracey): could use internal asm sum routine.
+	var v float64
+	for i := 0; i < s.mat.N; i++ {
+		v += s.mat.Data[i*s.mat.Stride+i]
+	}
+	return v
+}
+
 // GrowSym returns the receiver expanded by n rows and n columns. If the
 // dimensions of the expanded matrix are outside the capacity of the receiver
 // a new allocation is made, otherwise not. Note that the receiver itself is
diff --git a/mat/triangular.go b/mat/triangular.go
index a98cf3a3..e32ee405 100644
--- a/mat/triangular.go
+++ b/mat/triangular.go
@@ -217,6 +217,18 @@ func (t *TriDense) RawTriangular() blas64.Triangular {
 	return t.mat
 }
 
+// SetRawTriangular sets the underlying blas64.Triangular used by the receiver.
+// Changes to elements in the receiver following the call will be reflected
+// in the input.
+//
+// The supplied Triangular must not use blas.Unit storage format.
+func (t *TriDense) SetRawTriangular(mat blas64.Triangular) {
+	if mat.Diag == blas.Unit {
+		panic("mat: cannot set TriDense with Unit storage format")
+	}
+	t.mat = mat
+}
+
 // Reset zeros the dimensions of the matrix so that it can be reused as the
 // receiver of a dimensionally restricted operation.
 //
@@ -513,6 +525,16 @@ func (t *TriDense) ScaleTri(f float64, a Triangular) {
 	}
 }
 
+// Trace returns the trace of the matrix.
+func (t *TriDense) Trace() float64 {
+	// TODO(btracey): could use internal asm sum routine.
+	var v float64
+	for i := 0; i < t.mat.N; i++ {
+		v += t.mat.Data[i*t.mat.Stride+i]
+	}
+	return v
+}
+
 // copySymIntoTriangle copies a symmetric matrix into a TriDense
 func copySymIntoTriangle(t *TriDense, s Symmetric) {
 	n, upper := t.Triangle()
diff --git a/mat/triband.go b/mat/triband.go
index 71ad38ff..f9785504 100644
--- a/mat/triband.go
+++ b/mat/triband.go
@@ -321,6 +321,18 @@ func (t *TriBandDense) RawTriBand() blas64.TriangularBand {
 	return t.mat
 }
 
+// SetRawTriBand sets the underlying blas64.TriangularBand used by the receiver.
+// Changes to elements in the receiver following the call will be reflected
+// in the input.
+//
+// The supplied TriangularBand must not use blas.Unit storage format.
+func (t *TriBandDense) SetRawTriBand(mat blas64.TriangularBand) {
+	if mat.Diag == blas.Unit {
+		panic("mat: cannot set TriBand with Unit storage")
+	}
+	t.mat = mat
+}
+
 // DiagView returns the diagonal as a matrix backed by the original data.
 func (t *TriBandDense) DiagView() Diagonal {
 	if t.mat.Diag == blas.Unit {