diff --git a/lapack/testlapack/dtrexc.go b/lapack/testlapack/dtrexc.go
index 4b20675e..59d0d3d0 100644
--- a/lapack/testlapack/dtrexc.go
+++ b/lapack/testlapack/dtrexc.go
@@ -27,7 +27,7 @@ func DtrexcTest(t *testing.T, impl Dtrexcer) {
 		for _, n := range []int{1, 2, 3, 4, 5, 6, 10, 18, 31, 53} {
 			for _, extra := range []int{0, 1, 11} {
 				for cas := 0; cas < 100; cas++ {
-					tmat := randomSchurCanonical(n, n+extra, rnd)
+					tmat, _, _ := randomSchurCanonical(n, n+extra, rnd)
 					ifst := rnd.Intn(n)
 					ilst := rnd.Intn(n)
 					testDtrexc(t, impl, compq, tmat, ifst, ilst, extra, rnd)
@@ -38,7 +38,7 @@ func DtrexcTest(t *testing.T, impl Dtrexcer) {
 
 	for _, compq := range []lapack.UpdateSchurComp{lapack.UpdateSchurNone, lapack.UpdateSchur} {
 		for _, extra := range []int{0, 1, 11} {
-			tmat := randomSchurCanonical(0, extra, rnd)
+			tmat, _, _ := randomSchurCanonical(0, extra, rnd)
 			testDtrexc(t, impl, compq, tmat, 0, 0, extra, rnd)
 		}
 	}
diff --git a/lapack/testlapack/general.go b/lapack/testlapack/general.go
index 075a4006..96a16f54 100644
--- a/lapack/testlapack/general.go
+++ b/lapack/testlapack/general.go
@@ -131,8 +131,8 @@ func randomHessenberg(n, stride int, rnd *rand.Rand) blas64.General {
 // form, that is, block upper triangular with 1×1 and 2×2 diagonal blocks where
 // each 2×2 diagonal block has its diagonal elements equal and its off-diagonal
 // elements of opposite sign.
-func randomSchurCanonical(n, stride int, rnd *rand.Rand) blas64.General {
-	t := randomGeneral(n, n, stride, rnd)
+func randomSchurCanonical(n, stride int, rnd *rand.Rand) (t blas64.General, wr, wi []float64) {
+	t = randomGeneral(n, n, stride, rnd)
 	// Zero out the lower triangle.
 	for i := 0; i < t.Rows; i++ {
 		for j := 0; j < i; j++ {
@@ -141,23 +141,49 @@ func randomSchurCanonical(n, stride int, rnd *rand.Rand) blas64.General {
 	}
 	// Randomly create 2×2 diagonal blocks.
 	for i := 0; i < t.Rows; {
-		if i == t.Rows-1 || rnd.Float64() < 0.5 {
+		p := rnd.Float64()
+		var r float64
+		switch {
+		case p < 0.5:
+			// A half of real parts of eigenvalues will be "normal".
+			r = t.Data[i*t.Stride+i]
+		case p < 0.75:
+			// A quarter will be very small.
+			r = dlamchS
+		default:
+			// A quarter will be zero.
+		}
+		// Store the actual value back at the diagonal of T.
+		t.Data[i*t.Stride+i] = r
+
+		// A half of eigenvalues will be real.
+		if rnd.Float64() < 0.5 || i == t.Rows-1 {
 			// 1×1 block.
+			wr = append(wr, r)
+			wi = append(wi, 0)
 			i++
 			continue
 		}
+
 		// 2×2 block.
 		// Diagonal elements equal.
-		t.Data[(i+1)*t.Stride+i+1] = t.Data[i*t.Stride+i]
+		t.Data[(i+1)*t.Stride+i+1] = r
+		// Element under diagonal very small or "normal".
+		c := dlamchS
+		if rnd.Float64() < 0.5 {
+			c = rnd.NormFloat64()
+		}
 		// Off-diagonal elements of opposite sign.
-		c := rnd.NormFloat64()
 		if math.Signbit(c) == math.Signbit(t.Data[i*t.Stride+i+1]) {
 			c *= -1
 		}
 		t.Data[(i+1)*t.Stride+i] = c
+		wr = append(wr, r, r)
+		c = math.Sqrt(math.Abs(t.Data[i*t.Stride+i+1])) * math.Sqrt(math.Abs(c))
+		wi = append(wi, c, -c)
 		i += 2
 	}
-	return t
+	return t, wr, wi
 }
 
 // blockedUpperTriGeneral returns a normal random, general matrix in the form